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On Distributed and Acoustic Sensing for Situational Awareness On Distributed and Acoustic Sensing for Situational Awareness

FANGRONG PENG Syracuse University

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ABSTRACT

Recent advances in electronics enable the development of small-sized, low-cost, low-

power, multi-functional sensor nodes that possess local processing capability as well as to

work collaboratively through communications. They are able to sense, collect, and process

data from the surrounding environment locally. Collaboration among the nodes are enabled

due to their integrated communication capability. Such a system, generally referred to as

sensor networks are widely used in various of areas, such as environmental monitoring,

asset tracking, indoor navigation, etc.

This thesis consists of two separate applications of such mobile sensors. In this first

part, we study decentralized inference problems with dependent observations in wireless

sensor networks. Two separate problems are addressed in this part: one pertaining to col-

laborative spectrum sensing while the other on distributed parameter estimation with cor-

related additive Gaussian noise. In the second part, we employ a single acoustic sensor

with co-located microphone and loudspeaker to reconstruct a 2-D convex polygonal room

shape.

For spectrum sensing, we study the optimality of energy detection that has been widely

used in the literature. This thesis studies the potential optimality (or sub-optimality) of

the energy detector in spectrum sensing. With a single sensing node, we show that the

energy detector is provably optimal for most cases and for the case when it is not theoret-

ically optimal, its performance is nearly indistinguishable from the true optimal detector.

For cooperative spectrum sensing where multiple nodes are employed, we use a recently

proposed framework for distributed detection with dependent observations to establish the

optimality of energy detector for several cooperative spectrum sensing systems and point

out difficulties for the remaining cases.

The second problem in decentralized inference studied in this thesis is to investigate

the impact of noise correlation on decentralized estimation performance. For a tandem

network with correlated additive Gaussian noises, we establish that threshold quantizer on

local observations is optimal in the sense of maximizing Fisher information at the fusion

center; this is true despite the fact that subsequent estimators may differ at the fusion cen-

ter, depending on the statistical distribution of the parameter to be estimated. In addition,

it is always beneficial to have the better sensor (i.e. the one with higher signal-to-noise

ratio) serve as the fusion center in a tandem network for all correlation regimes. Finally,

we identify different correlation regimes in terms of their impact on the estimation per-

formance. These include the well known case where negatively correlated noises benefit

estimation performance as it facilitates noise cancellation, as well as two distinct regimes

with positively correlated noises compared with that of the independent case.

In the second part of this thesis, a practical problem of room shape reconstruction using

first-order acoustic echoes is explored. Specifically, a single mobile node, with co-located

loudspeaker, microphone and internal motion sensors, is deployed and times of arrival of

the first-order echoes are measured and used to recover room shape. Two separate cases

are studied: the first assumes no knowledge about the sensor trajectory, and the second one

assumes partial knowledge on the sensor movement. For either case, the uniqueness of

the mapping between the first-order echoes and the room geometry is discussed. Without

any trajectory information, we show that first-order echoes are sufficient to recover 2-D

room shapes for all convex polygons with the exception of parallelograms. Algorithmic

procedure is developed to eliminate the higher-order echoes among the collected echoes

in order to retrieve the room geometry. In the second case, the mapping is proved for any

convex polygonal shapes when partial trajectory information from internal motion sensors

is available. A practical algorithm for room reconstruction in the presence of noise and

higher order echoes is proposed.

ON DISTRIBUTED AND ACOUSTIC SENSING FOR

SITUATIONAL AWARENESS

By

Fangrong PengB. S., Huazhong University of Science and Technology, Wuhan, China, 2008

M. S., Chalmers University of Technology, Gothenburg, Sweden, 2010

DISSERTATION

Submitted in partial fulfillment of the requirements for the degree ofDoctor of Philosophy in Electrical and Computer Engineering

Syracuse UniversityAugust 2016

Copyright c© 2016 Fangrong Peng

All rights reserved

ACKNOWLEDGMENTS

First of all, I would like to express my gratitude to my advisor Dr. Biao Chen for his

continued and unreserved supports during my doctoral studies, both intellectually

and financially. It was a great pleasure to be a student of his. I sincerely thank

him for his guidance on my research with his intellect, motivation, encouragement

and patience, and the meetings with him makes me think more thoroughly and

rigorously on my research topics. Without his help, I could not have finished this

long and tough journey. I also want to thank Mrs. Chen for taking care of everyone

in our lab, and she will always be deeply in my heart.

I would like to take this opportunity to thank my doctoral committee (alphabet-

ically): Prof. Hao Chen, Prof. Pinyuen Chen, Prof. Yingbin Liang, Prof. Jian Tang,

and Prof. Yanzhi Wang. Special thanks go to Dr. Pramod Vashney and Dr. Makan

Fardad, who are unfortunately not on my committee due to time conflict but pro-

vided valuable suggestions and encouragements to my defense. I would also like

to thank my labmate Tiexing Wang for his collaboration in obtaining the results

in Chapter 5 and Chapter 6. Thanks to Dr. Henk Wymeersch for leading me into

the research world and providing me initial confidence. I am greatly indebted to

my Communication Lab fellows, including Xiaohu, Yi, Wei, Minna, Ge, Fangfang,

Kapil, Pengfei, Yu, Yang and Tiexing, with whom I shared my joy, and many even

become my lifetime friends.

As a third-generation Ph.D. in my family, my parents thoroughly understand

how tough the Ph.D. life is, and they offer me unconditional supports in every

v

aspect of my life. Special thanks to my father Dr. Fang Peng, for his inspiring

advices when I was stuck in a mathematical proof. The thesis also goes in memory

to my dearest grandmother, who passed away when I was absent from her, which

left a deep hole in my heart permanently. I am also thankful to my beloved half Dr.

Sijia Liu, with whom I go through my Ph.D. years also as friends and colleagues,

and who makes me a better person.

I am grateful for my best friends in the United States, Dr. Yuting Zhang, Jiujiu

Sun and Si Chen, with whom I began the relationships even back to high school,

and from whom I still can get charged every time we hug each other. I am also

obliged to my friends Yunyun Bi and Yue Duan for their genuine help during a

hard time of my life.

Thanks to everyone who ever helped me and encouraged me, and who ever

questioned me and challenged me.

vi

To my family:

past, present, and future.

TABLE OF CONTENTS

Acknowledgments v

List of Tables xi

List of Figures xii

1 Introduction 1

1.1 Decentralized Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Energy Detection in Spectrum Sensing . . . . . . . . . . . . . . . 2

1.1.2 Data Dependency and Redundancy in a Tandem Network . . . . . . 3

1.2 Room Shape Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Decentralized Inference with Dependent Observations 6

2.1 Bayesian Distributed Detection . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Hierarchical Conditional Independence Model . . . . . . . . . . . . . . . . 10

2.2.1 CHCI model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Energy Detection in Cooperative Spectrum Sensing 13

3.1 Spectrum Sensing With A Single Node . . . . . . . . . . . . . . . . . . . . 14

3.1.1 Gaussian Signal over AWGN Channel . . . . . . . . . . . . . . . . 16

3.1.2 PSK Signal over Rayleigh Fading Channel . . . . . . . . . . . . . 16

3.1.3 QAM Signal over Rayleigh Fading Channel . . . . . . . . . . . . . 18

3.2 Cooperative Spectrum Sensing . . . . . . . . . . . . . . . . . . . . . . . . 21

viii

3.2.1 Gaussian Signal over AWGN channel . . . . . . . . . . . . . . . . 24

3.2.2 PSK Signal over Fast Rayleigh Fading Channel . . . . . . . . . . . 25

3.2.3 QAM Signal over Fast Rayleigh Fading Channel . . . . . . . . . . 26

3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Decentralized Estimation with Correlated Observations in a Tandem Network 28

4.1 Fisher Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.1 Classical Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.2 Bayesian Framework . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Centralized Estimation under Gaussian Noise . . . . . . . . . . . . . . . . 34

4.3 Decentralized Estimation with Bivariate Gaussian Noises . . . . . . . . . . 38

4.3.1 The Optimality of Single Threshold Quantizer . . . . . . . . . . . 38

4.3.2 Communication Direction Problem . . . . . . . . . . . . . . . . . 44

4.3.3 Data Dependency and Redundancy . . . . . . . . . . . . . . . . . 46

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5 Room Shape Recovery via a Single Mobile Acoustic Sensor without Trajectory

Information 53

5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.1.1 Image Source Model . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.1.2 Mapping between ISM and RIR . . . . . . . . . . . . . . . . . . . 54

5.2 Room Shape Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.2.1 Identifiability by First-Order TOAs . . . . . . . . . . . . . . . . . 56

5.2.2 Algorithm for Triangle Reconstruction with Only First-order TOAs 57

5.2.3 Polygon Reconstruction by First-Order TOAs . . . . . . . . . . . . 58

5.2.4 Algorithm of Polygon Reconstruction with Higher-order TOAs . . 60

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

ix

6 Room Shape Recovery via a Single Mobile Acoustic Sensor with Trajectory

Information 66

6.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.1.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.2 Recovery with Known Distances and Unknown Path Direction . . . . . . . 69

6.3 Experimental result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.3.1 Experiment Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.3.2 Signal Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.3.3 Room Shape Reconstruction Experiment . . . . . . . . . . . . . . 74

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7 Conclusion and Future Research Directions 77

7.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7.2 Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Appendices 80

A Distribution of PSK and QAM signals over Rayleigh fading channel 80

B Proof of Lemma 4.3 82

C Proof of Lemma 4.2 87

D Centralized Estimation for a Multi-Node System in Gaussian Noises 90

E Proof of Lemma 5.1 93

References 96

x

LIST OF TABLES

xi

LIST OF FIGURES

2.1 a canonical distributed detection system . . . . . . . . . . . . . . . . . . . 7

3.1 ROC curves of the LRT and energy detector for 16QAM in Rayleigh fad-

ing. The SNR, defined in dB as 10 log10

¯r2mσ2

σ2w

, is at 5dB, and r2m is the

average energy of QAM signals. . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Deflection of two detectors within corresponding SNR ranges . . . . . . . . 22

4.1 Two fusion structures under a communication constraint where ψi(·) is as-

sumed to be a one-bit quantizer in the present chapter . . . . . . . . . . . . 29

4.2 Comparison between J(X; θ) and J(X; θ|Y ) . . . . . . . . . . . . . . . . 37

4.3 PCRLB vs. γ for different ρ. . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.4 PCRLB comparison with ρ = −0.5, 0, 0.5 . . . . . . . . . . . . . . . . . . . . . . 46

4.5 Comparison between J(U(X); θ) and J(U(X); θ|Y ) . . . . . . . . . . . . 47

4.6 CRLB, its bounds and MSE of the Decentralized Estimation Problem . . . 49

4.7 MSE vs. γ for different ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.8 Estimation MSE comparison with ρ = −0.5, 0, 0.5 . . . . . . . . . . . . . 51

4.9 Estimation MSE with different SNR at the worse node . . . . . . . . . . . 51

5.1 Image Source Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2 original (blue) and recovered (dash purple/black) polygons: (a) an irregular

quadrilateral, (b) a rotation, (c) a reflection, (d) a hexagon . . . . . . . . . . 64

xii

6.1 A mobile device is employed to measure the geometry of a room. The

mobile device collects echoes atO1,O2 andO3 successively. The distances

between the consecutive measurement points are d12 and d23 . . . . . . . . 68

6.2 Correlator output at O1 towards the first wall. Peaks with solid ellipses

correspond to true walls. Peaks with dash ellipses correspond to either

noise or higher-order echoes . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.3 Comparison between the ground truth (black underlined) and experiment

result (red) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

E.1 (a) one side is tilted, (b) two sides are tilted . . . . . . . . . . . . . . . . . 94

xiii

1

CHAPTER 1

INTRODUCTION

This thesis consists of two parts. The first part focuses on decentralized inference with the

emphasis on problems involving dependent observations across sensors. The second part

deals with a practical research problem, in which we proposed computer-aided algorithms

to reconstruct the shape of a room by using a single mobile acoustic sensor.

1.1 Decentralized Inference

Decentralized inference refers to the decision making process in a system where multiple

distributed sensors are involved [1]. In such a system, each sensor observes a phenomenon

of interest and transmits a compressed version of its observation to a fusion center (FC).

Meanwhile, the FC makes final decision regarding the phenomenon using the information

collected from the peripheral sensors, as well as its own observation. This general topic has

drawn extensive attention over the past decades [2–6]. However, most of those works con-

sider the scenario that the observations across sensors are statistically independent. When

dependence occurs, the problem becomes much more challenging, as the local decision

rule or quantizer design is usually coupled with each other. In this part of the thesis, we

investigate two problems. The first one deals with decentralized detection; specifically, the

2

objective is to detect the presence of transmitted signals from the so-called primary user in

the context of spectrum sensing using multiple nodes. The second one is on decentralized

estimation where multiple sensors collectively estimate a parameter of interest. A common

thread for both problems are that the observations across sensors are correlated with each

other, both for the spectrum sensing and the decentralized estimation problems.

1.1.1 Energy Detection in Spectrum Sensing

We first study the problem of decentralized detection. Specifically, we analyze the energy

detection in a cooperative spectrum sensing system in terms of its potential optimality. The

spectrum sensing problem comes from the rising demand for accessing the wireless spec-

trums with the dramatically increasing number of wireless devices. Coincidentally, it has

also long been recognized that there is gross under-utilization in licensed radio frequency

bands [7]. In this case, dynamic spectrum access (DSA) has been proposed as one solution

to resolve the spectrum crunch [8], which allows multiple secondary users to access spec-

trum space whenever the designated primary user is idle. A key enabling technology for

DSA is spectrum sensing, i.e., a secondary user should only communicate when it believes

that the primary user is indeed silent to avoid unintended interference to the primary user.

However, many factors, including path-loss, shadowing, and channel fading, can degrade

the sensing performance when only a single node is used for spectrum sensing. For exam-

ple, the location of the secondary user may hinder its ability to hear the primary transmitter.

To overcome this difficulty, cooperative spectrum sensing is introduced where spatially dis-

tributed nodes collaboratively detect the presence of the primary user’s signal [9, 10].

While there have been numerous studies in spectrum sensing, it is almost without ex-

ception that an energy detector is used for both stand-alone spectrum sensing and coop-

erative spectrum sensing systems. It is somewhat surprising that there has not been any

systematic study about the suitability or optimality of the energy detector for various spec-

trum sensing applications. Particularly, with random primary signals which are common

3

for all sensors, observations are correlated across sensing nodes given the hypothesis that

the primary user is active. It is well known that for dependent observations, designing

optimal local decision rules is generally an NP hard problem due to the coupling effect

across sensors. This thesis fills this gap by analyzing energy detection in both stand-alone

and cooperative spectrum sensing systems, thus providing a clear guidance on when such

a detector may indeed be optimal for spectrum sensing.

1.1.2 Data Dependency and Redundancy in a Tandem Network

We then examine a problem within decentralized estimation system. The objective of dis-

tributed estimation is to estimate at the FC an underlying parameter which is indicative of

the phenomenon of interest. Extensive studies have been reported for this topic during the

decades (see [5,6,11–13] and references therein), and many fundamental results have been

obtained. However, most results assume that the observations are conditionally indepen-

dent (CI). We consider the challenging problem of decentralized estimation with dependent

observations.

A decentralized estimation system usually requires quantization prior to the communi-

cations between the local sensors and the FC. This is due to various system limitations and

resource constraints that collectively impose a finite capacity constraint. In the absence of

CI assumption, the local quantizer design problem parallels that of local decision rule in

decentralized detection system, which was known to be a NP hard problem. The work in

this thesis provides some new insight on the optimum quantizer design for decentralized

estimation with dependent observations. A better understanding of the optimum quantizer

structure will in turn enable us to answer some of the interesting and important questions

arising in decentralized inference. Two of them addressed in this dissertation include: op-

timal sensing architecture in terms of communication direction and the characterization of

the correction regimes in terms of its impact on the estimation performance.

4

1.2 Room Shape Reconstruction

In the second part of this thesis, we study the indoor room shape recovery problem using

a mobile acoustic sensor. Specifically, assuming a convex polygonal room and a mobile

sensor with co-located loudspeaker and microphone, we study the problem of reconstruct-

ing the 2-D room geometry using the acoustic response of the room measured at multiple

locations.

While reliable and accurate outdoor localization can be obtained by using primarily the

global positioning system (GPS), significant obstacles exist to achieve similar localization

performance in an indoor environment. GPS signals, due to its high frequencies, are often

unavailable in an indoor environment due to severe attenuation, blockage, and other impair-

ments. Instead, existing indoor localization often relies on pre-existing infrastructure such

as WiFi signals, Bluetooth, ultra-wide band (UWB), LED light. For example, the tech-

nology introduced by WiFiSLAM (Simultaneous Localization and Mapping) requires the

full coverage of WiFi signals for the indoor environment as well as pre-existing 2-D maps

for learning and localization. Similarly, for UWB, LED light or Bluetooth based systems,

certain anchor nodes need to be fixed in advance at some known locations in the indoor

environment.

There are situations that the infrastructure is either unavailable or inaccessible. Even

with applications where infrastructure may be pre-established, they may not be available

in the event of natural disaster as power outage may render the infrastructure inaccessible.

Therefore, being able to reconstruct the geometry of the surrounding environment and self-

localize in the absence of pre-established infrastructure is critical for applications such as

rescue missions by first responders.

Indoor room shape reconstruction using acoustic sensors has been an active field of

research in recent years [14–19]. Most existing works, however, assume multiple loud-

speaker and/or distributed microphones that simultaneously transmit and receive echoes

within the room to be examined. In [14], a single loudspeaker and a microphone array

5

were used to measure multiple single wall impulse response in different angles. Then an

l-1 regularized least squares method was applied to map the measured impulse responses

to a 3D shoebox room. The number of walls was assumed to be known in this model.

In [15], the time of arrival (TOA) was measured by a set of microphones, and an algo-

rithm to eliminate the higher-order reflective signals (signals bouncing over more than one

obstacles) and estimate the geometry was proposed, based on an interesting property of

the Euclidean distance matrix. Without any prior information, especially the number of

walls and the order of reflections, an TOA-based method was introduced in [16], where

the TOAs were estimated by the generalized correlation method. A different model was

proposed in [17], where multiple acoustic stimuli were used to generate echoes reflected

under different angles to a single microphone. The reconstruction problem was addressed

by finding common tangents of ellipses.

The work that is most closely related to our work is that of [18], where a single co-

located loudspeaker and microphone is used for room shape recovery. Both first-order and

second-order echoes are utilized in [18]. It was pointed out in [18] that first-order echoes

alone are not sufficient for even the simplest room shape such as a triangle with a static

sensor. This thesis introduces mobility of the acoustic sensor where multiple measurement

points are used to collect acoustic echoes. Under minor assumption on the the measurement

point (e.g., they are on a straight line) we establish that 2-D room shape reconstruction is

indeed feasible using first order acoustic echoes when supplemental geometry information

is available through internal motion sensor.

6

CHAPTER 2

DECENTRALIZED INFERENCE WITH

DEPENDENT OBSERVATIONS

The first part of this thesis focus on decentralized inference with an emphasis on the prob-

lems containing dependent observations. In this chapter, we review a recently proposed

technique tackling dependent observations [20] in the canonical distributed detection sys-

tem as shown in Fig. 2.1.

In a canonical distributed detection network, each sensor observes a phenomenon dis-

tributed according to p(x1, · · · , xK |H), and then makes a decision regarding the hypothesis

H, then the local decisions U1, · · · , UK are transmitted to the FC, where a final decision

U0 is made based on the received local decisions. Different from the centralized detection,

the information accessible at the FC is the quantized version instead of the entirety of the

original observations, usually due to the bandwidth limit. Extensive work has been done

on this problem that leads to many fundamental results [2–4, 21–24].

In this system, there are two different design problems, the fusion rule and the local

decision rule. The optimum fusion rule at the FC is known to be the likelihood ratio test

(LRT) [22–24], while designing the local decision rule is much more complicated due to

the coupling effect in the distributed setting. Most works assume that the local observations

7

p(x1, · · · , xK |H)

X1

XK

Sensor 1γ1(·)

Sensor KγK(·)

...

U1

UK

Fusion Centerγ0(U1, · · · , UK)

U0

Fig. 2.1: a canonical distributed detection system

are conditionally independent of each other given each hypothesis, which implies that the

joint distribution of the observations can be decomposed as

p(x1, · · · , xK |Hi) =K∏k=1

p(xk|Hm), m = 0, · · · ,M − 1, (2.1)

where M is the number of hypotheses in the detection problem.

Under this assumption, the problem can be significantly simplified. It is shown that

the likelihood quantizer is optimal for the local decision rule [4, 25], while the coupling

among the distributed sensors reduces to that on choosing the thresholds. In [2], a person-

by-person optimization method is proposed to find the optimal quantization threshold for

each local sensor.

In the case that the distributed sensors detect a random signal with independent noises

or a deterministic signal embedded in correlated noises, the conditional independence (CI)

assumption is violated, and the problem becomes generally intractable. It is shown in [26]

that the local decision rule design is a NP complete problem when the observations are

conditionally dependent, and the likelihood quantizer is generally not optimal, even for the

relatively simple problem of two sensors observing a shift in mean correlated Gaussian

random variables [27, 28].

8

A unifying model under the Bayesian inference framework is proposed in [20], and

both conditionally independent and dependent observations can be considered as special

cases of the new model. In such model, a hidden random variable W is introduced such

that the observations are conditionally independent on the new random variable, even if in

the original model they are conditionally dependent given the underlying hypothesis. The

new framework drastically simplifies the local decision rule design issue in a wide classes

of distributed detection with dependent observations. The framework is described in the

following section.

2.1 Bayesian Distributed Detection

Consider a parallel distributed hypothesis testing system withM hypotheses andK sensors

illustrated in Fig. 2.1. The joint distribution of the local observations under each hypothesis

p(x1, · · · , xK |H), H ∈ {0, · · · ,M − 1} is assumed to be known. Each sensor makes its

local decision regarding the hypotheses based on its own observation Xk, and transmit the

local decision Uk = γk(Xk) ∈ {0, · · · ,M − 1} to the FC. Finally, the global decision

U0 = γ0(U1, · · · , UK) ∈ {0, · · · ,M − 1} is made by the FC. Let the prior distribution

of the hypothesis H denoted as πH , and the observations written in a vector form as X =

{X1, · · · , XK}. For simplicity, we denote {X1, · · · , Xk−1, Xk+1, XK} as Xk.

Generally, in the parallel distributed system, there is

p(u|x) =K∏k=1

p(uk|xk), (2.2)

and the following Markov chain satisfies

H −X−U− U0. (2.3)

The goal in this parallel distributed detection system is to minimize the expected Bayesian

9

cost by choosing a set of decision rules {γ0, · · · , γK}. Let cu0,h be the Bayesian cost of

deciding U0 = u0 when H = h is true, and for measuring the probability of error, cu0,h

takes value from the set {0, 1}. Then the Bayesian cost of the whole system is

C =M−1∑u0=0

M−1∑h=0

cu0,hπhp(u0|h)

=

∫X

∑u

M−1∑u0=0

M−1∑h=0

cu0,hπhp(u0|u)p(u|x)p(x|h)dx, (2.4)

where (2.4) follows from the Markov chain H −X−U− U0. Then expand the Bayesian

cost C with respect to the k-th sensor, we have

C =

∫Xk

∑uk

p(uk|xk)fk(uk, xk)dxk, (2.5)

where fk(uk, xk) is the Bayesian cost density function (BCDF) for the kth sensor making

decision uk while observing xk, and it is defined as

fk(uk, xk) ,∑uk

M−1∑u0=0

M−1∑h=0

cu0,hπhp(u0|uk, uk)∫Xk

p(uk|xk)p(xk, xk|h)dxk

=M−1∑u0=0

M−1∑h=0

cu0,hπhp(xk|h)p(u0|uk, xk, h), (2.6)

where (2.7) follows because

p(u0|uk, xk, h) =∑uk

p(u0|uk, uk)∫Xk

p(uk|xk)p(xk|xk, h)dxk.

From (2.5) one can see that to minimize the expected Bayesian cost, the k-th local sensor

need to make a decision uk such that fk(uk, xk) is minimized given fixed fusion rule and

local decision rule at other sensors. On the other hand, since fk(uk, xk) is coupled with

the fusion rule and other sensor’s decision rule, finding the optimal fusion rule for the k-th

sensor is generally intractable.

10

In the case that the observations are conditionally independent, the joint distribution

can be decomposed according to (2.1), which leads the BCDF to

fk(uk, xk) ,∑uk

M−1∑u0=0

M−1∑h=0

cu0,hπhp(u0|uk, uk)∫Xk

p(uk|xk)p(xk|h)p(xk|h)dxk

=M−1∑u0=0

M−1∑h=0

cu0,hπhp(xk|h)p(u0|uk, h),

=M−1∑h=0

αk(uk, h)p(xk|h). (2.7)

Therefore, the optimal decision rule γk(Xk) reduces to an optimal M -ary Bayesian hy-

potheses test, i.e.

Uk = γk(Xk) = arg minuk

M−1∑h=0

αk(uk, h)p(xk|h).

2.2 Hierarchical Conditional Independence Model

A hierarchical conditional independence (HCI) model [20] introduces a new random vari-

able W such that the following two conditions are satisfied,

1. the following Markov chain holds

H −W −X−U− U0.

2. X1, · · · , XK are conditionally independent given W, i.e.

p(x1, · · · , xK |W) =K∏k=1

p(xk|W).

11

Under this assumption, the joint distribution of local observations under each hypothesis

can be rewritten as

p(x1, · · · , xK |H) =

∫W

p(w|H)K∏k=1

p(xk|w). (2.8)

Notice that the “hidden” variable can be scalar or vector. It has been shown that any dis-

tributed detection problem characterized by (2.3) can be represented by the HCI model [20].

The HCI model can be classified into three categories according to the support of W, while

in this chapter, we only focus on the continuous HCI (CHCI) model.

2.2.1 CHCI model

In this model, the conditional independence of local observations given the new random

variable enables us to rewrite the BCDF:

fk(uk, xk) =

∫W

M−1∑u0=0

M−1∑h=0

cu0,hπhp(u0|uk, w)p(w|h)p(xk|w)dw

=

∫W

βuk,wp(xk|w)dw. (2.9)

It is shown in [20] that by imposing additional constraints on W , a class of CHCI model

can be determined where the optimal local decision rules are the threshold quantizers on

local observations. The conclusion is repeated in the following proposition.

Proposition 2.1. [20, Proposition 1] Consider a distributed detection system with binary

hypothesis, binary sensor outputs, and the sensor observations are scalars. Suppose that

the distributed detection problem can be described equivalently by the CHCI model where

W is a scalar, and the following three conditions are satisfied:

1. The fusion center implements a monotone fusion rule, i.e.

P (U0 = 1|Uk = 1, w) ≥ P (U0 = 1|Uk = 0, w);

12

2. The ratio p(w|H1)p(w|H0)

is a nondecreasing function of w;

3. The ratio p(xk|w)

p(x′k|w)

is also a nondecreasing function of w for any xk > x′k.

Then there exists a single threshold quantizer at the kth sensor

Uk =

1 xk ≥ τk;

0 xk < τk,

that minimizes the error probability at the fusion center.

Proposition 2.1 serves as a new method to dealing with some distributed detection prob-

lems with conditionally dependent observations which seem formidable. The examples in-

clude the two shift in mean correlated Gaussian random variables [27, 28]. However, this

proposition only works for the case that the sensor observations are scalars, and it will be

seen in the next chapter that with vector observations, one needs to properly transform the

observations before this proposition can be applied.

13

CHAPTER 3

ENERGY DETECTION IN COOPERATIVE

SPECTRUM SENSING

The Energy detector was originally proposed for the detection of an unknown deterministic

signal in additive white Gaussian noise (AWGN) [29]. The performance of energy detector

of unknown deterministic signals over fading channels was analyzed in [30]. However, for

spectrum sensing, it is often more convenient, and perhaps more accurate to treat the signals

from primary user as random. Indeed, such assumption, i.e., digital signals are considered

random is the very premise upon which the theory of optimum receiver in digital communi-

cations is developed. Additionally, the fading channels naturally randomize the transmitted

signals even if one considers the signals themselves as deterministic. The assumption that

the primary user’s signal is of a random nature allows us to study the optimality of the

energy detector.

In a spectrum sensing system, it keeps monitoring the spectrum occupancy to ensure

the secondary users access the sparse portion without causing any undue interference to

the primary user. In such a system, energy detection is heuristically used without validated

optimality. The simplest model is to assume that the signal itself is Gaussian; if the the

channel itself is also assumed to be AWGN, an energy detector is provably optimal and

14

its performance can be analyzed in a straightforward manner [31]. The problem becomes

much more difficult for cooperative spectrum sensing, even under the simple assumption of

Gaussian signal over Gaussian channel. The presence of the common signal from the pri-

mary user introduces dependence among observations at different nodes and decentralized

detection with dependent observations is a perennially difficult problem [28].

The chapter includes the detailed studies of the optimality or sub-optimality of the en-

ergy detector for various spectrum sensing systems. For single node spectrum sensing, we

show that energy detection is provably optimal for detecting phase shift keying (PSK) sig-

nals in Rayleigh fading channels. For detecting quadrature amplitude modulation (QAM)

in Rayleigh fading channels, energy detection is provably optimal for the low and high

SNR regimes; for moderate SNR, while it is strictly sub-optimal, we show through nu-

merical examples that its performance is practically identical to the optimum detector. For

cooperative spectrum sensing, considerable challenges exist to obtain the optimality for the

most general model. However, we are able to establish the optimality of energy detection

for the following cases: Gaussian signals in Gaussian channels with a single observation;

PSK signal in independent Rayleigh fading channels; QAM in independent Rayleigh fading

channels with a single observation. Our results rely on the technique of tackling decentral-

ized detection with dependent observations introduced in Chapter 2.

3.1 Spectrum Sensing With A Single Node

Consider the following hypothesis testing (HT) problem: for n = 1, · · · , N ,

H0 : y(n) = w(n)

H1 : y(n) = x(n) + w(n), (3.1)

where n stands for the time index, and w(n) is assumed throughout this chapter to be inde-

pendent and identically distributed (i.i.d.) according to CN (0, σ2w), i.e., complex Gaussian

15

with zero mean and variance σ2w. One can equivalently formulate the problem as testing

whether or not x(n) = 0. The canonical form of an energy detector is the following test.

N∑n=1

|y(n)|2H1

≷H0

λ, (3.2)

for some specified threshold λ. The energy detector was first proposed in [29] where x(n)

is assumed to be an unknown but deterministic signal. The detection performance can be

derived by the distribution of the test statistic specified below.

N∑n=1

|y(n)|2 ∼

σ2w

2χ2

2N H0;

σ2w

2χ2

2N(2snr) H1,

where 2snr is the non-centrality parameter of the non-central chi-squared distribution, and

snr =∑Nn=1 |x(n)|2σ2w

is the signal to noise ratio (SNR) which depends on the total signal

energy. Subsequently, the probability of detection (Pd) and probability of false alarm (Pf )

can be obtained as

Pd = P

{N∑n=1

|y(n)|2 > λ | H0

}

= QN

(√

2snr,

√λ

σ2w/2

),

Pf = P

{N∑n=1

|y(n)|2 > λ | H1

}

=Γ(N, λ

σ2w

)Γ(N)

,

where QN (·, ·) is the generalized Marcum Q-function [32], representing the complement

of the cumulative density function (CDF) of a non-central chi-squared distributed random

variable, while Γ(·) is the gamma function, and Γ(·, ·) is the incomplete gamma function.

The use of energy detection in detecting unknown deterministic signals in fading chan-

16

nel was further studied in [30] and the detection performance was analyzed under various

fading scenarios.

In the following sections, we will instead consider x(n) as a random signal, as is cus-

tomary in the context of digital communications. We begin with the simple Gaussian signal

model.

3.1.1 Gaussian Signal over AWGN Channel

By assuming that x(n) is i.i.d. complex Gaussian [31] with zero mean and variance σ2s , the

HT problem in (3.1) becomes a classical random signal detection problem for which the

energy detector can be easily shown to be the optimum LRT [33]. The distributions of the

test statistic∑N

n=1 |y(n)|2 can be easily obtained to be [31]

N∑n=0

|y(n)|2 ∼

σ2w

2χ2

2N H0;

σ2w+σ2

s

2χ2

2N H1.

Thus, instead of a test between a central chi-squared distribution against a non-central

chi-squared distribution as in the case with deterministic signals, the problem becomes a

test of two central chi-squared distributions with identical degrees of freedom but different

scalings. The corresponding Pd and Pf are respectively

Pd =Γ(N, λ

σ2w+σ2

s)

Γ(N),

Pf =Γ(N, λ

σ2w

)

Γ(N). (3.3)

3.1.2 PSK Signal over Rayleigh Fading Channel

Instead of AWGN channels, consider now that the primary signal is subject to a fading

channel. For simplicity, we only consider Rayleigh flat fast fading in this thesis. As such,

17

the complex baseband signal in the HT problem (3.1) becomes

x(n) = s(n)h(n),

where s(n) is a complex baseband signal while the fading coefficient h(n) are zero-mean

complex Gaussian variables with variance σ2 whose amplitude follows a Rayleigh distri-

bution. Furthermore, h(n) are assumed to be independent of each other for different n, i.e.,

a fast fading channel. Let S be the set of signal constellations for the primary user, the HT

problem is to test the following two hypotheses: for n = 1, · · · , N,

H0 : y(n) = w(n);

H1 : y(n) = s(n)h(n) + w(n). (3.4)

In this subsection, we consider PSK modulation withM constellation points as the primary

user’s signal i.e. s ∈ S, where

S =

{sm : sm = ejθm , θm = m · 2π

Mfor m = 1, · · · ,M

}(3.5)

For simplicity, sm is assumed to be of unit energy as the signal energy can be easily ab-

sorbed into the fading channel statistics. Now let πm be the prior probability for sm, thus∑Mm=1 πm = 1. We show that x(n) is also a complex Gaussian random variable. For

ease of notation, we use p(x) to denote px(n)(x) in the following and similar for joint and

conditional distributions. The distribution of the signal x(n) is represented by

p(x) =∑m

p(x, sm)

=∑m

πmp(x|sm)

=∑m

πmp(h · sm)

18

Given that h is circularly symmetric complex Gaussian distributed and that sm = ejθm ,

the product h · sm has exactly the same distribution as h for every m (rigorous proof see

Appendix A ). Thus the above mixture of densities gives rise to exactly the same distribu-

tion as h itself. It further implies that the signal x(n) = h(n)s(n) observed at the sensing

node is also a complex Gaussian signal. Since x(n) is an independent Gaussian sequence

across time where the independence comes from the independent fast fading assumption,∑Nn=1 |y(n)|2 is again the LRT for the HT problem. Hence an energy detector is optimal

for the detection of PSK signals in independent Rayleigh fading channels.

3.1.3 QAM Signal over Rayleigh Fading Channel

Suppose instead that the set S in (3.5) comes from a QAM constellation. Thus S is defined

as

S ={sm : sm = rme

jθm , for m = 1, · · · ,M}, (3.6)

where rm is not identical for all m and comes from a finite set related to the quantity M ,

while θm corresponds to each rm. Again assign πm as the prior probability to sm. Similar

to that for PSK, straightforward derivation shows that the density of x(n) is a mixture of

Gaussian densities but with different variances

p(y(n)|H1) =M∑m=1

πm ·1

π(r2mσ

2 + σ2w)e− |y(n)|2

r2mσ2+σ2w

Apparently, it does not reduce further to a Gaussian variable as in the PSK case. Neverthe-

less, for a single sample, we can write out the likelihood ratio to be:

LR(y(n)) =

∑Mm=1 πm · 1

π(r2mσ2+σ2

w)· e−

|y(n)|2

r2mσ2+σ2w

1πσ2

w· e−

|y(n)|2σ2w

=M∑m=1

πm · σ2w

r2mσ

2 + σ2w

· er2mσ

2

σ2w(r2mσ2+σ2w)

|y(n)|2, (3.7)

19

which is clearly a monotone increasing function of |y(n)|2. Thus, if only a single sample

(i.e., N = 1) is considered instead of a long sequence, the energy detector is again proved

to be optimal.

For the cases of N > 1, under the independent fading assumption, the LR for N sam-

ples becomes:

LR(y) =N∏n=1

M∑m=1

πm · σ2w

r2mσ

2 + σ2w

· er2mσ

2

σ2w(r2mσ2+σ2w)

·|y(n)|2. (3.8)

Apparently, the LR is not in general a monotone increasing function of∑N

n=1 |y(n)|2.

Therefore, no general optimality of the energy detector can be claimed for such a case.

However, we consider different regimes of SNR and show in the following that the energy

detector is indeed asymptotically optimal in the high and low SNR regimes. The SNR in

such fast fading channel is given by r2mσ2

σ2w

.

1. High SNR regime.

Supposer2mσ

2

σ2w

→∞. (3.9)

Thus (3.8) can be approximated as

LR(y) ≈N∏n=1

M∑m=1

πmσ2w

r2mσ

2 + σ2w

e|y(n)|2

σ2w

=

(M∑m=1

πmσ2w

r2mσ

2 + σ2w

)N N∏n=1

e|y(n)|2

σ2w

=

(M∑m=1

πmσ2w

r2mσ

2 + σ2w

)N

e1

σ2w·∑Nn=1 |y(n)|2

,

which is apparently a monotonically increasing function of the signal energy∑N

n=1 |y(n)|2.

Therefore, the energy detector is asymptotically optimal under the high SNR regime.

2. Low SNR regime.

20

Suppose insteadr2mσ

2

σ2w

→ 0. (3.10)

With this assumption, (3.8) can be approximated as:

LR(y) ≈N∏n=1

M∑m=1

πmer2mσ

2

σ4w·|y(n)|2

≈N∏n=1

M∑m=1

πm

(1 +

r2mσ

2

σ4w

· |y(n)|2)

=N∏n=1

(1 +

M∑m=1

πmr2mσ

2

σ4w

· |y(n)|2). (3.11)

where we use the approximation ex ≈ 1 + x for small x. Taking logarithm of (3.11)

on both sides, the Log-Likelihood Ratio (LLR) is:

LLR(y) =N∑n=1

log

(1 +

M∑m=1

πmr2mσ

2

σ4w

· |y(n)|2)

≈N∑n=1

M∑m=1

πmr2mσ

2

σ4w

· |y(n)|2

=M∑m=1

πmr2mσ

2

σ4w

·N∑n=1

|y(n)|2

where the approximation follows from log(1 + x) ≈ x for small x. Again, for low

SNR, the LRT reduces approximately to the energy detector.

3. Moderate SNR.

Besides of the two extreme cases discussed above, however, the energy detector is

not provably optimal. Figs. 3.1, 3.2(a) and 3.2(b) are numerical examples comparing

the LRT with that of the energy detector: the receiver operating characteristic (ROC)

curve and the deflection measures [33] for the two test statistics. Fig. 3.1 is plotted at

SNR=5 while Figs. 3.2(a) and 3.2(b) plot the deflection measures at different SNRs.

In Fig. 3.2(a) the difference of the deflection measures between the two detectors

21

Pf

10-5

10-4

10-3

10-2

10-1

100

Pd

0.9994

0.9995

0.9996

0.9997

0.9998

0.9999

1

LRTEnergy

Fig. 3.1: ROC curves of the LRT and energy detector for 16QAM in Rayleighfading. The SNR, defined in dB as 10 log10

¯r2mσ2

σ2w

, is at 5dB, and r2m is the

average energy of QAM signals.

apprears negligible; however, if we zoom in the curve in a partinuclar interval, one

can see noticeable difference in Fig. 3.2(b). LRT does outperform the energy detector

albeit the difference is still small.

Thus for practical purposes, energy detector can be considered optimal for the QAM

in Rayleigh fading case. From (3.8), the reason that the energy detector is nearly

optimal can be attributed to the fact that the summation of different exponential terms

is dominated by those points with large rm. Thus the effect of unequal variances

disappears if only the dominating terms are kept.

Summarizing, for single node spectrum sensing, energy detector is either provably op-

timal or close to optimal for all the cases investigated in this section.

3.2 Cooperative Spectrum Sensing

We now turn our attention to the case where spectrum sensing involves multiple nodes,

i.e., cooperative spectrum sensing. Clearly, the problem of cooperative spectrum sensing

22

SNR-10 -8 -6 -4 -2 0 2 4 6 8 10

Deflection(d

B)

-10

-5

0

5

10

15

20

25

30

35

40

LRT

Energy

SNR5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Deflection(d

B)

24

26

28

30

32

34

36

LRT

Energy

(a) Deflection of two detectors for SNR from −10 to 10dB(b) Deflection of two detectors for SNR from 5 to 10dB

Fig. 3.2: Deflection of two detectors within corresponding SNR ranges

becomes a classical decentralized detection problem. Suppose there are a total of K sen-

sors (or nodes) in the system, cooperative spectrum sensing attempts to detect whether the

primary user’s signal is present or absent, i.e., to distinguish the following two hypotheses:

for k = 1, · · · , K,

H0 : yk(n) = wk(n),

H1 : yk(n) = xk(n) + wk(n), (3.12)

where xk(n) is the received signal at node k from the primary user and wk(n) is i.i.d. (in

n) complex Gaussian noise for each k. Each sensor is assumed to make a binary decision,

denoted by Uk, and report it to the fusion center where the final decision is made using

U1, · · · , UK .

As with any decentralized detection problems, there are two different design issues: the

fusion rule design and the decision rules at distributed sensors. The fusion rule design is

a rather straightforward exercise given that it has input from all the sensors: if the fusion

rule knows the local sensor decision rule, then optimal LRT can be easily implemented

at the fusion center. The difficulty lies in the design of local decision rules as they are

coupled with each other. Such coupling is especially acute when sensor observations are

23

conditionally dependent given one or both hypotheses. For such general cases, the optimal

local sensor decision rule design is an NP-complete problem [26]. With conditionally

independent observations, on the other hand, LRTs at local sensors are optimal for both the

Bayesian and NP frameworks, and the coupling is only in terms of that the thresholds of

the LRTs at different sensors are coupled [4, 34]. This significantly simplifies the problem

as one can iteratively update, for each sensor, the LRT threshold; locally optimal thresholds

can thus be easily attained.

For cooperative spectrum sensing, the corresponding decentralized detection problem

has to deal with dependent observations: signals received at different sensors are generated

by the same primary user, thus xk(n) are typically dependent of each other for different

k under H1. As such, even in cases where the energy detector coincides with the LRT at

individual node, there is no guarantee that the energy detector is also optimal for the coop-

erative spectrum sensing system. In the following, we study several scenarios that parallel

those cases studied in the previous section. Our approach in establishing the optimality of

the energy detector for some of the cases largely relies on the results obtained in [35]. The

new framework introduced in [35] assumes the existence of a hidden variable that induces

conditional independence among otherwise dependent observations. Provided that the hid-

den variable satisfies certain conditions, then the optimal local sensor decision rules can be

obtained in a straightforward manner. We repeat below Proposition 2.1 (also Proposition 1

in [35]) which is key to our approach. We denote y(n) , [y1(n), y2(n), · · · , yK(n)]T and

tailor the proposition to our problem.

Proposition 3.1. If there exists a scalar random variableW such thatH−W−y(n) forms

a Markov chain, and it satisfies the following conditions:

1. The fusion center implements a monotone fusion rule;

2. The ratio p(w|H1)p(w|H0)

is a nondecreasing function of w;

3. The ratio p(yk(n)|w)p(y′k(n)|w)

is also a nondecreasing function of w for any yk(n) > y′k(n).

24

Then there exists a single threshold quantizer at the kth sensor

Uk =

1 yk(n) ≥ τk;

0 yk(n) < τk,

that minimizes the error probability at the fusion center.

We discuss in the following the Gaussian signal over the AWGN channel and the PSK

as well as QAM under Rayleigh fading channel in the cooperative spectrum sensing system.

3.2.1 Gaussian Signal over AWGN channel

Given the model in (3.12), suppose xk(n) = x(n), i.e., all sensors observe an identical

signal and furthermore, assume x(n) is itself Gaussian with variance σ2s . Thus the HT

problem becomes a test between two multivariate Gaussian distributions

H0 : y(n) ∼ N (0, σ2wI),

H1 : y(n) ∼ N (0,C),

where

C =

σ2s + σ2

w σ2s · · · σ2

s

σ2s σ2

s + σ2w · · · σ2

s

... . . . . . . σ2s

σ2s σ2

s · · · σ2s + σ2

w

.

Notice that this example is a slight generalization for the example described in [35, Section

V.C] which deals with real variables. The proof of optimality of a threshold test of |y(n)|2

carries over to the complex case. However, the result applies only to the case whereN = 1.

Notice that here the optimality is weaker than that for a single node with the same signal

model; there, the energy detector is optimal for all N .

25

3.2.2 PSK Signal over Fast Rayleigh Fading Channel

Consider the model in (3.12) with xk(n) = hk(n)s(n) and hk(n) is a complex Gaussian

variable and is independent in k and n, s(n) is drawn from a PSK constellation set S

defined in (3.5). From Section 3.1, we know that xk(n) is itself a complex Gaussian random

variable. We show in the following that these xk(n) are indeed independent of each other

for different k. Or more generally,

Lemma 3.1. For PSK signals over Rayleigh fading channels, if the PSK sequence from the

primary user is independent in time and the fading channels are independent across both

k (sensors) and n (time), then the received signal vectors yk = [yk(1), · · · , yk(N)] are

independent across different sensors given either hypothesis.

Proof. Under H0, the independence is trivial since the noise variables are assumed inde-

pendent across both time n and sensor k. UnderH1, yk(n) = hk(n)s(n)+wk(n). However,

it was shown in Section 3.1 that xk(n) = hk(n)s(n) is complex Gaussian distributed. We

now show that for any k 6= k′, xk(n) and xk′(n′) are independent of each other for any n

and n′. For n 6= n′, the independence is trivially true as (hk(n), s(n)) and (hk′(n′), s(n′))

are independent of each other. For n = n′, we show that xk(n) and xk′(n) are independent

Gaussian distributed random variables. 1

Following similar approach in establishing the Gaussian distribution of xk(n), we have

p(xk, xk′) =M∑m=1

p(xk, xk′ , sm)

=M∑m=1

p(xk, xk′ |sm)πm

(a)=

M∑m=1

p(xk|sm)p(xk′ |sm)πm

(b)= phk(xk)phk′ (xk′)

1Notice that proving uncorrelatedness between xk(n) and xk′(n) is not sufficient; uncorrelated Gaussianrandom variable does not necessarily imply independence unless the two are jointly Gaussian.

26

where (a) follows as xk(n) and xk′(n) are conditionally independent given s(n) and (b)

is because given s(n) = sm, xk and xk′ respectively have the exact distribution of hk(n)

and hk′(n). Thus we have shown that xk(n) and xk′(n) are independent Gaussian random

variables.

Now that we have proved that the observations yk are conditionally independent across

k given either hypothesis, local LRT at each sensor is optimal [4, 34] . Given that the LRT

is equivalent to an energy detector for PSK in independent Rayleigh fading channels, as

was established in Section 3.1, the energy detector is therefore also optimal for cooperative

spectrum sensing.

3.2.3 QAM Signal over Fast Rayleigh Fading Channel

For QAM signal in Rayleigh fading channels, it was already established that the energy

detector is not optimal in the single node case for the general case of N > 1. Thus for the

distributed case, we only consider the simple case ofN = 1, i.e., using a single observation.

From (3.8), it is clear that givenH1, yk(n)’s are not independent for different k.

We now use the approach in [35] to show that a threshold test of |yk(n)|2 is still optimal

forN = 1. As we only consider one sample, in the following we suppress the time index n.

Let us first define a hidden variable such that the observations become independent given

the hidden variable. Define

W = |s|I(H = H1) =

0 H = H0

|s| H = H1,

where I(H = H1) is an indicator function. It is clear thatH−W−y forms a Markov chain,

i.e., given W , the distribution of y is independent ofH. To see this, if W = 0, then y is the

noise vector w, thus is independent of H. Conditioned on W = |sm|, we know that this is

equivalent to sending a PSK symbol with signal energy |sm|2. The distribution y is again

27

dependent only on |sm| given the circular invariance of complex Gaussian. Additionally,

W induces conditional independence of y, again, for the same reason as using PSK, the

observations are conditionally independent. Thus W serves as the hidden variable in the

framework described in [35]. However, Proposition 3.1 does not directly apply here given

that the observations are complex (i.e., the monotone property defined for yk > yk′ is not

applicable).

Nevertheless, an alternative proof can be constructed. First, given W = |s|, yk is a

circularly invariant complex Gaussian random variable. As such, W − |yk| − yk forms a

Markov chain, i.e., given |yk|, yk is independent of W . Given that yk’s are conditionally

independent given W , (|y1|, · · · , |yK |) forms a sufficient statistic for W [36]. Thus they

also form a sufficient statistic forH. Now that we can instead consider |yk|’s as local obser-

vations, it is then a straightforward exercise to check that the conditions in Proposition 3.1

is satisfied with the defined W . As such, a threshold test of |yk|, or equivalently |yk|2 is

optimal for the cooperative sensing problem with QAM in Rayleigh fading channels and

N = 1.

3.3 Summary

This chapter studied the optimality of an energy detector for spectrum sensing for both a

stand-alone system and a cooperative spectrum sensing system. For the single node case,

it was shown that for all the cases considered in this chapter, the energy detector is either

provably optimal or nearly optimal compared with the true optimum test. For coopera-

tive spectrum sensing, however, optimality was established only for several special cases.

These include PSK over independent Rayleigh fading for arbitrary sample numbers, and

the special case of Gaussian signals in Gaussian channels and QAM signals in Rayleigh

fading channels when the detector is limited to using a single sample.

28

CHAPTER 4

DECENTRALIZED ESTIMATION WITH

CORRELATED OBSERVATIONS IN A

TANDEM NETWORK

Consider the simplest estimation model with two distributed sensors collecting noisy ob-

servations of a parameter θ:

X = θ +W1, Y = θ +W2, (4.1)

where (W1,W2) ∼ N (0, 0, σ21, σ

22, ρ), i.e., the noises are bivariate normal distributed with

zero mean, respective variances σ21, σ

22 , and correlation coefficient ρ. Without loss of gen-

erality, σ21 ≥ σ2

2 is assumed throughout this chapter unless otherwise stated. Of particular

interest to the present chapter is a tandem network, where one node serves as the FC, and

makes decisions based on its own observation as well as the compressed (quantized) input

received from the other node. Such a two-node tandem network is illustrated in Fig. 4.1.

With Gaussian model, a natural criterion for evaluating the estimation performance is the

mean squared error (MSE). With distributed estimation and quantized observations, com-

29

puting MSE is often cumbersome and even intractable. Instead, Cramer-Rao lower bound

(CRLB) [37,38] is often used which is equivalent to evaluating the Fisher information (FI)

given the observation model. Thus we will primarily use FI in assessing distributed estima-

tors with dependent observations; as CRLB is not always tight with quantized observations,

we will also resort to MSE evaluation when feasible.

p(x, y|θ)

X Y

θ1 = f1(U, Y )

U = ψ1(X)

p(x, y|θ)

X Y

θ2 = f2(X, V )

V = ψ2(Y )

(a) X to Y tandem network (b) Y to X tandem network

Fig. 4.1: Two fusion structures under a communication constraint where ψi(·)is assumed to be a one-bit quantizer in the present chapter

Local quantizer design in a distributed estimation system has been examined earlier in,

e.g., [5] [39], where the focus is largely on algorithmic design and the accompanied nu-

merical results. More concrete analytical results have been obtained in [40] [41], where

score-function quantizer (SFQ) is shown to be optimal for maximizing the Fisher infor-

mation (FI). The result, however, is derived under the assumption that observations are

conditionally independent.

In the absence of the CI assumption, decentralized inference becomes much more chal-

lenging [42, 43]. In this case, the structure of the optimal quantizer at local sensors is

usually coupled with other nodes. This difficulty is much well understood for distributed

detection with dependent observations. Under the CI assumption, likelihood-ratio quan-

tizers (LRQ) at local sensors have been shown to be optimal under both the Bayesian

and Neyman-Pearson frameworks [44]. They are also known to maximize the Kullback-

Leibler distance and Chernoff information for distributed detection, again, under the CI

30

assumption [45]. With this assumption removed, LRQs at local sensors are often not opti-

mal [46,47], and the quantizer design becomes NP hard for the general dependent case [45].

The problem of interest to the present chapter, namely decentralized estimation with depen-

dent observations has yet to receive much attention. An early work [39] presented some

numerical results of threshold quantizer design with dependent observations, though no

analytical result has been obtained.

By obtaining and understanding the optimum quantizer design for decentralized esti-

mation with dependent observations, we are able to answer some of the interesting and

important questions arising in decentralized inference. We address two of them in this

chapter: 1) what is the preferred communication direction in tandem networks so that the

ultimate inference performance at the FC is optimized, and 2) how does data dependency

influence the inference performance compared with that of the independent case?

The communication direction problem can be illustrated using the simple two-node tan-

dem network. Figs. 4.1(a) and (b) represent two different configurations where either X

or Y serves as the FC; the question is which one yields better performance given a joint

distribution p(x, y). This issue was first introduced and analyzed in distributed detection

with CI assumption under either hypothesis [48]. In this early work, the optimal configura-

tion is proved to be dependent on external factors, such as prior probabilities of hypotheses

and cost assignments. The authors in [49] addressed a similar problem under the additive

correlated Gaussian noise model; by restricting the peripheral nodes to implement LRQ,

the authors established that the preferred communication direction depends on the corre-

lation across sensor observations. More recently, for the same model considered in [49],

the optimal decision structure is obtained for the peripheral node in [50], and it was shown

that with additive Gaussian noises, having the better sensor (i.e., the one with higher SNR)

serving as the FC is always preferred regardless of the correlation coefficient.

The ultimate goal in this section is to measure the impact of data correlation on the

inference performance. As mentioned before, data dependency is usually believed to bring

31

in data redundancy as it reduces the effective sample size [33, Ch. 5]. However, there are

counter-examples showing that data correlation can be exploited to significantly benefit the

inference performance, such as the noise canceling effect in the case of negatively corre-

lated noises in centralized estimation problems [51, Ch. 6]. Compared to the centralized

setup, the problem of how data correlation affects estimation in decentralized systems is

much more complicated, and it is one of the problems of interest in this chapter.

4.1 Fisher Information

4.1.1 Classical Setting

In an inference problem, an observationX carries information about an unknown parameter

θ, which the probability of X depends on, and that amount of information is characterized

by FI. The likelihood function of θ is the probability ofX conditioned on the value of θ, and

the score function, denoted as Sθ(x) is defined as the partial derivative of the log-likelihood

function with respect to θ,

Sθ(x) =∂

∂θlog p(x|θ).

In classical statistics, the FI is defined as the expectation of the second moment of the score,

J(X; θ) = E

[(∂

∂θlog p(x|θ)

)2]

= E[S2θ (x)

].

The inverse of FI, J(X; θ)−1, namely CRLB, serves as a lower bound on the variance

of estimators of a deterministic parameter. Particularly, for any unbiased estimator, the

estimation MSE is at least as high as the FI, and the one which achieves the CRLB is an

efficient estimator. Nonetheless, there may not exist an unbiased estimator which attains

the bound. The CRLB can be also used to bound the variance of biased estimators given

32

the bias, and it is possible that some biased estimators yield variance and MSE lower than

the CRLB.

4.1.2 Bayesian Framework

In the Bayesian framework, the unknown parameter θ is treated as a random variable.

Therefore, the information carried by the observation X about the parameter can not be

subject to a particular value of θ, and the parameter itself carries a certain amount of prior

information which has to be incorporated. The Bayesian Fisher information 1 is defined

as [37]

JB(X; θ) = EX,Θ

[(∂

∂θlog p(x, θ)

)2]

, JD(X; θ) + JP (θ)

if the expectation exists. The first term

JD(X; θ) = EΘEX|θ

[(∂

∂θlog p(x|θ)

)2]

is the FI associated with data averaged over θ, and the second term

JP (θ) = EΘ

[(∂

∂θlog p(θ)

)2]

is the prior information associated with p(θ). Similar to the classical case, the PCRLB,

defined as JB(X; θ)−1, provides a theoretical lower bound for the estimation MSE of any

estimator under the Bayesian framework.

As with other meaningfully defined information quantities, such as mutual information,

1Note that there are different names for Fisher information when prior information is incorporated, e.g.,Bayesian information in [37]. We will simply refer to them as Fisher information which takes a form thatincorporate prior distribution whenever applicable.

33

the averaged FI satisfies the chain rule [52], i.e.,

JD(X, Y ; θ) = JD(Y ; θ) + JD(X; θ|Y ),

where the conditional FI is defined as

JD(X; θ|Y ) = EΘEX,Y |θ

[(∂

∂θlog p(x|y, θ)

)2].

Therefore, the joint Bayesian FI can be decomposed as

JB(X, Y ; θ) = JD(Y ; θ) + JD(X; θ|Y ) + JP (θ). (4.2)

Notice that with independent Gaussian noises in (4.1), the joint likelihood function factor-

izes into the product of the marginal distributions, leading to

JB(X, Y ; θ) = JD(Y ; θ) + JD(X; θ) + JP (θ). (4.3)

Clearly, from (4.2) and (4.3), the effect of data correlation on the estimation performance

amounts to comparing JD(X; θ|Y ) and JD(X; θ), provided that there exists an estimator

that attains the PCRLB. Redundancy occurs when JD(X; θ|Y ) < JD(X; θ), i.e., FI de-

creases with conditioning. For the sake of simplicity, the subscript of JD(·) is dropped

without any ambiguity.

34

4.2 Centralized Estimation under Gaussian Noise

Consider the estimation problem in the bivariate Gaussian model described in (4.1). It is

straightforward to show that, with |ρ| < 1

J(X, Y ; θ) =1

1− ρ2·[

1

σ21

+1

σ22

− 2ρ

σ1σ2

], (4.4)

J(X; θ) =1

σ21

, (4.5)

J(Y ; θ) =1

σ22

. (4.6)

For the case with |ρ| = 1, the observations are perfectly correlated, indicating the degen-

erate cases of bivariate Gaussian, so the corresponding FI can be obtained as taking the

appropriate limit of J(X, Y ; θ) when ρ → ±1, where the limit is defined in the usual

sense.

To examine the effect of data correlation, we now compare J(X; θ) and J(X; θ|Y ) for

the Gaussian additive model. From (4.4), (4.5), and (4.2), we get

J(X; θ|Y ) =1

1− ρ2·(ρ

σ2

− 1

σ1

)2

. (4.7)

Comparing (4.7) with (4.5), we can categorize various correlation regimes in terms of in-

ference performance relative to that of the independence case.

• ρ = −1. As X and Y are perfectly and negatively correlated (i.e., σ2W1 = −σ1W2),

complete noise cancellation can be achieved by linearly combining X and Y . Notice

that in the additive Gaussian model, the optimal (MMSE) estimator happens to be

linear. The optimal estimator for θ is

θ =σ2

σ1 + σ2

x+σ1

σ1 + σ2

y,

which can be shown to be identically θ hence the MSE is 0. Equivalently, we obtain

35

the limiting FI when ρ→ −1,

J(X; θ|Y ) = limρ→−1

( 1σ1− ρ

σ2)2

1− ρ2→∞,

which is consistent with the fact that perfect estimation can be achieved with ρ = −1.

• −1 < ρ < 0. For this regime, it is straightforward to show that partial noise cancel-

lation is optimal in the sense of minimizing the MSE:

θ =σ2

2 − ρσ1σ2

σ22 + σ2

1 − 2ρσ1σ2

x+σ2

1 − ρσ1σ2

σ22 + σ2

1 − 2ρσ1σ2

y

with the corresponding MSE (1−ρ2)·σ21σ

22

σ21+σ2

2−2ρσ1σ2. Equivalently, the FI for this case can be

shown to be greater than that of the independent case, i.e.,

J(X; θ|Y )− J(X; θ) =

ρ2

σ21− 2ρ

σ1σ2+ ρ2

σ22

1− ρ2> 0.

Thus, negatively correlated Gaussian noises in the additive model (4.1) is always

beneficial for the estimation performance. Notice that in the centralized case Gaus-

sian model, the minimum mean squared error (MMSE) estimator - coincides with the

linear minimum mean squared error (LMMSE) estimator. This is not the case with

the decentralized case where the peripheral node needs to quantize its observation.

• 0 ≤ ρ ≤ 2σ1σ2σ21+σ2

2. This is when dependency implies redundancy, i.e.,

J(X; θ|Y )− J(X; θ) =ρ2(

1σ21

+ 1σ22

)− 2ρ

σ1σ2

1− ρ2≤ 0.

Thus data dependence negatively affects the estimation performance compared with

that of the independent case. At the particular point ρ = σ2σ1

, the conditional FI be-

comes 0, implying that X is completely redundant given Y . This is the consequence

36

of the following Markov chain θ − Y −X when ρ = σ2σ1

, i.e., X is independent of θ

given Y .

• 2σ1σ2σ21+σ2

2< ρ < 1. This is the parameter regime where positively correlated noises also

benefit the inference performance. Checking the FI, it is easy to show that for this

parameter regime,

J(X; θ|Y )− J(X; θ) =ρ2(

1σ21

+ 1σ22

)− 2ρ

σ1σ2

1− ρ2> 0.

To understand why this is the case, we note that for positively correlated noises,

(partial) noise cancellation is also attainable by subtracting one observation from the

other with proper scaling. However, the subtraction also reduces the signal power.

The balancing point happens to be at

ρ =2σ1σ2

σ21 + σ2

2

, (4.8)

i.e., beyond this value, noise cancellation more than compensates for the signal power

reduction, resulting in improved estimation performance.

• ρ = 1. This is the extreme case when σ2W1 = σ1W2. Depending on the values of σ1

and σ2, there are two distinct cases that have completely different ramifications on

the underlying estimation problem.

– σ1 = σ2, i.e., W1 = W2, thus X = Y . Therefore one of the observations is

completely redundant. The conditional FI is now

J(X; θ|Y ) =1

σ21

limρ→1

(1− ρ)2

1− ρ2= 0.

Indeed, this is exactly a special case of the extreme point corresponding to

ρ = σ2σ1

hence X is redundant given Y . In this case, 2σ1σ2σ21+σ2

2= 1 hence the

37

ρ-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

Fis

he

r In

form

atio

n

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

J(X;θ)J(X;θ|Y)

σ1

2 = 4, σ

2

2 = 1

Fig. 4.2: Comparison between J(X; θ) and J(X; θ|Y )

previous parameter regime 2σ1σ2σ21+σ2

2< ρ < 1 vanishes.

– σ1 > σ2. In this case, perfect estimation is achieved by the following estimator

θ =σ1

σ1 − σ2

y − σ2

σ1 − σ2

x.

Not surprisingly, the corresponding FI becomes unbounded as that when ρ =

−1 and perfect estimation is also achieved.

J(X; θ|Y ) = limρ→−1

( 1σ1− 1

σ2)2

1− ρ2→∞.

The FI of the various parameter regimes are plotted in Fig. 4.2 where the noise variances

are fixed as σ1 = 2 and σ2 = 1. With this setting, we can compute the boundary point

(4.8) to be ρ = 0.8. From the figure, it is clear that J(X; θ|Y ) ≥ J(X; θ) for ρ ∈ [−1, 0]

and ρ ∈ [0.8, 1], while the opposite inequality holds for ρ ∈ (0, 0.8). Additionally, the

conditional FI J(X; θ|Y ) = 0 at ρ = 12

= σ2σ1

indicating that X is completely redundant

given Y .

38

4.3 Decentralized Estimation with Bivariate Gaussian

Noises

We now proceed to consider the problem when X or Y is subject to quantization prior

to being available at the other node that estimates θ. The requirement is often times due

to various system constraints which collectively impose a finite capacity constraint for the

communication between the two nodes. For either of the two configurations in Fig. 4.1,

we first address the optimal quantizer design at the remote node. Subsequently, we attempt

to answer the question of which node should be used as a FC for better estimation per-

formance. Finally, the resulting quantizer structure as well as the optimal communication

direction will reveal how correlation may impact the estimation performance.

4.3.1 The Optimality of Single Threshold Quantizer

For simplicity, we assume the extreme case of a one-bit quantizer at the local sensor. Con-

sider Fig. 4.1(a). The FI at the estimator decomposes into three terms

J(Y, U(X); θ) = J(Y ; θ) + J(U(X); θ|Y ) + J(θ),

where U(X) is the binary quantizer output for X . Hence maximizing the overall FI

J(Y, U(X); θ) is equivalent to maximizing the conditional term J(U(X); θ|Y ). A special

case is when the noises are independent. In this case, the second term becomes independent

of Y , i.e., one only need to design a quantizer at X such that the FI of the quantizer output

is maximized. The dependent case is more complicated since Y is not accessible at node

X , therefore it is not realistic to design a quantizer to maximize FI for each specific value

of Y . However, we will show later that the class of optimal quantizer is in fact independent

of Y .

The definition of the score-function quantizer (SFQ) is given in [40].

39

Definition 4.1. A quantizerψ(·) is a monotone SFQ with threshold vector t = (t1, · · · , tD−1) ∈

RD−1, if ψ(x) = d⇔ Sθ(x) ∈ [td−1, td] for any x, where t0 = −∞ ≤ t1 ≤ · · · ≤ tD−1 ≤

∞ = tD. Any permutation of the monotone SFQ generated by a permutation mapping

π : {1, · · · , D} → {1, · · · , D} is a SFQ.

The significance of SFQ is that it optimizes the FI J(U(X); θ) among all quantizers

with the same quantization level. In a tandem network, to maximize the conditional FI

J(U(X); θ|Y ), the corresponding notion of the score-function is characterized by the con-

ditional distribution as

Sθ(x|y) =∂

∂θlog p(x|θ, y), (4.9)

which is dependent on both θ and y. Therefore, we denote the SFQ corresponding to

Sθ(x|y) as ψθ,y(x) for the time being. The reason that SFQ ψθ,y(x) can not directly apply

to our case is two-fold: 1) Score-function is dependent on the parameter θ, whose value is

unknown a priori and it is generally not possible to design a single quantizer that is optimal

for every θ [41]; and 2) in a tandem network, the observation from the other node (Y ) is not

available at the quantizer node (X). However, by defining the class of SFQ’s at a particular

pair (θ, y) as Ψθ,y = {ψθ,y(x) : ψθ,y(x) is a SFQ at (θ, y)}, and then by using a similar

argument as in [41], it can be shown that if the conditional distribution p(x|θ, y) satisfies

a monotonicity property, then the class of SFQ’s at every single pair of (θ, y) is identical.

This is summarized in the following lemma.

Lemma 4.1. Let T (X) be a function of X . If the score function can be expressed as

Sθ(x|y) = fθ,y(T (x)), where fθ, y(·) is monotone increasing for any (θ, y), then

1. the class of SFQ’s, Ψθ,y, is identical for all (θ, y) pairs, i.e., for any (θ, y) and (θ′, y′),

Ψθ,y = Ψθ′,y′;

2. every SFQ is equivalent to a quantizer on T (x) with D−1 thresholds, i.e. there exist

40

−∞ = t′0 ≤ t′1 ≤ · · · ≤ t′D−1 ≤ t′D =∞ such that

ψ(x) = d⇔ T (x) ∈ [t′d−1, t′d].

Proof. Since Sθ(x|y) is monotone increasing in T (x), any SFQ is equivalent to quantizing

T (x) while retaining the order of thresholds. Therefore, the class of SFQ’s is independent

of θ and y.

For the problem of a tandem network with bivariate Gaussian noises, Sθ(x|y) is deter-

mined by

Sθ(x|y) =∂

∂θlog

1√2π(1− ρ2)σ2

1

e−

(x−θ−σ1σ2 ρ(y−θ))2

2(1−ρ2)σ21

=∂

∂θ

−(x− θ − σ1

σ2ρ(y − θ)

)2

2(1− ρ2)σ21

=

1− σ1σ2ρ

(1− ρ2)σ21

(x− σ1

σ2

ρy −(

1− σ1

σ2

ρ

)θ

),

It is clear that we can choose T (x) = x for −1 ≤ ρ < σ2σ1

, and T (x) = −x for σ2σ1< ρ ≤ 1,

such that Lemma 4.1 applies. Therefore, we can straightforwardly show that the optimal

one-bit quantizer in the Gaussian additive model is a single threshold quantizer on the

observation itself, i.e.

ψ(X) = 1{x ≥ γ}.

Remark: Conditional sufficient statistics can help to find the function T (X). We give the

definition as follows.

Definition 4.2. A statistic W (X) is a conditional sufficient statistic for θ conditioned on

Y , if the conditional distribution of the sample X given the value of W (X) and Y does not

depend on θ. [53].

41

The Neyman-Fisher factorization theorem can be generalized to characterize condi-

tional sufficient statistics: W (X) is a conditional statistic for θ given Y iff p(x, y|θ) =

gθ(W (x), y) · h(x, y) for all x, y and θ [53]. In this case, the score function is also a

function of W (X). Then if the score function is monotone on W (X), one can simply set

T (X) = W (X).

The optimal threshold γ∗ ∈ R is one such that

γ∗ = arg maxγ

J(ψ(X, γ); θ|Y ),

where we use the notation ψ(X, γ) to indicate the dependence of the quantizer output on

the threshold γ. By definition, the conditional FI is characterized as

J(ψ(X, γ); θ|Y )

= EX,Y,Θ[Sθ(ψ(X, γ)|Y )2

]=

∫Θ

f(θ)

∫Yg(y|θ)

∑u∈{0,1}

P (ψ(x, γ) = u|θ, y) Sθ(ψ(x, γ)|y)2 dydθ, (4.10)

where f(θ) is the pdf of θ, f(y|θ) is the conditional distribution of Y given θ, and Sθ(ψ(x, γ)|y)

is defined according to (4.9).

We provide the following theorem to identify the optimal threshold γ∗ for the one-bit

quantizer in the additive Gaussian model.

Theorem 4.1. Let θ ∼ N (µ, σ2) be a parameter with Gaussian distribution. Let X and

Y be noisy observations of θ with bivariate additive Gaussian noise. Then, for the tandem

system in Fig. 4.1(a), the optimal one-bit quantizer on X is a single threshold quantizer on

X , and the optimal threshold is γ∗ = µ, i.e., U(X) = ψ(X,µ).

Without loss of generality, we assume µ = 0. We begin with the following lemma to

prove Theorem 4.1.

42

Lemma 4.2. Define the function η(t) = e−t2

Q(t)(1−Q(t)). Then η(t) is a symmetric function

with respect to t = 0, while monotonically decreasing in t ∈ (0,∞), and

t∗ = arg maxt

η(t) = 0, η∗ = 4.

Proof. The detailed proof is given in Appendix C.

With the Gaussian assumption described in Theorem 4.1, conditional FI in (4.10) can

be expanded into

J(ψ(X, γ); θ|Y ) =

∫ ∞−∞

∫ ∞−∞

e−θ2

2σ2√2πσ2

· e− (y−θ)2

2σ22√2πσ2

2

·(ρσ1−σ2)2

2π(1−ρ2)σ21σ

22· e−(γ−θ−σ1σ2

ρ(y−θ)√

1−ρ2σ1

)2

Q

(γ−θ−σ1

σ2ρ(y−θ)√

1−ρ2σ1

)(1−Q

(γ−θ−σ1

σ2ρ(y−θ)√

1−ρ2σ1

)) dydθ, (4.11)

Observe that the integrand can be made symmetric for (y, θ) around (0, 0) by setting γ = 0

(c.f. Lemma 4.2). Then this intuitively attains the maximum and we give a formal proof

below.

Proof. With the assumption µ = 0, we only need to show

J(ψ(X, 0); θ|Y )− J(ψ(X, γ); θ|Y ) > 0, ∀γ 6= 0. (4.12)

We first rewrite the integral in (4.11) as

J(ψ(X, γ); θ|Y )w=y−θ∝

∫ ∞−∞

∫ ∞−∞

f(θ)g(w)η(βγ − βθ − αw) dwdθ,

where we denote α = ρ√1−ρ2σ2

, and β = 1√1−ρ2σ1

, and f(·) and g(·) as the pdf’s of θ and

43

Y |θ, respectively. Plug it back into the left-hand side of (4.12), we have

∫ ∞−∞

∫ ∞−∞

f(θ)g(w)η(−βθ − αω) dθdw −∫ ∞−∞

∫ ∞−∞

f(θ)g(w)η(βγ − βθ − αw) dθdw

(1)=

∫ ∞−∞

∫ ∞−∞

f(θ)g(w)η(βθ + αw) dθdw −∫ ∞−∞

∫ ∞−∞

f(θ)g(w)η(βγ − βθ − αw) dθdw

(2)=

∫ ∞−∞

∫ γ2−αw

β

−∞

(f(θ′)− f(γ − 2αw

β− θ′)

)g(w)

× (η(βθ′ + αw)− η(βγ − βθ′ − αw)) dθ′dw. (4.13)

The first equality follows from Lemma 4.2, and the second one follows by splitting the

integral with respect to θ at γ2− αw

β, and change of variable θ′ = γ − 2αw

β− θ.

Without loss of generality, we first consider the interval of ω such that γ2− αw

β> 0.

Since both f(·) and η(·) are even functions symmetric at 0 and monotone decreasing for

t > 0, we can infer that f(θ′) − f(γ − 2αwβ− θ′) and η(βθ′ + αw) − η(βγ − βθ′ − αw)

are both non-negative, as θ′ < γ2− αw

βin the inner integral; while for γ

2− αw

β< 0, they are

both non-positive. Moreover, since g(w) is always positive, then the integrand in (4.13) is

always non-negative. On the other hand, the integrand equals 0 in two cases: 1) γ = 0, and

2) γ2− αw

β= 0. For the first case, γ = 0 implies that η(βθ′+αw)− η(βγ−βθ′−αw) = 0

for any w, so the whole integral also equals 0. However, for the second case, the integrand

is equal to 0 only for the particular point w = γβ2α

, so the whole term integrates above 0

when γ 6= 0. Hence, (4.12) is proved, and thus the unique optimal threshold is γ∗ = 0.

Intuitively, since both f(θ) and g(ω) are even functions around 0 and monotone de-

creasing as the corresponding parameter deviates further away from 0 (bell-shaped), it is

clear that maximizing the integral in (4.13) would result in placing the function η(·) around

the origin. It is also clear from the above proof that the Gaussian assumption on the pa-

rameter θ in Theorem 4.1 can be generalized to an arbitrary distribution with symmetric

bell-shaped pdf.

The PCRLB at the FC with different correlation coefficients are given in Fig. 4.3. In

44

γ-5 -4 -3 -2 -1 0 1 2 3 4 5

PC

RLB

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

ρ = -0.98ρ = -0.5ρ = 0ρ = 0.5ρ = 0.8ρ = 0.98

σ1

2 = 4, σ

2

2 = 1

Fig. 4.3: PCRLB vs. γ for different ρ.

this figure, the parameter is standard normal distributed, and for different ρ, the threshold γ

varies from−5 to 5, which is a fairly large range compared to the signal and noise standard

deviations. One can easily observe from the figure that the PCRLB is always minimized at

γ = 0.

4.3.2 Communication Direction Problem

For a given joint distribution p(x, y), the communication direction problem is equivalent

to comparing the performance between the configurations in Fig. 4.1(a) and (b). Our

goal is therefore to determine, for σ21 > σ2

2 , which of the two FIs J(U(X), Y ; θ) and

J(X, V (Y ); θ) is larger and whether the inequality depends on ρ, the correlation coeffi-

cient of the additive noises. Here U(X) and V (Y ) are respectively the optimal quantizer

outputs at X and Y , which are both single threshold quantizers at γ∗ = 0.

For any arbitrary threshold γ ∈ R, we can establish the following inequality,

Lemma 4.3. Let ψ(·, γ) be a single threshold quantizer with threshold γ. For the observa-

45

tions X and Y given in (4.1) and with σ21 > σ2

2 , we have

J(ψ(X, γ), Y ; θ) > J(X,ψ(Y, γ); θ), ∀γ ∈ R (4.14)

Proof. See Appendix B.

In the last section, we showed that the optimal thresholds for quantizing X and Y

are both at γ∗ = 0 for zero-mean Gaussian parameters. We show in the following that

Lemma 4.3 implies that the preferred direction is independent of the prior distribution on

θ. Indeed, it is true even if the parameter is treated as an unknown deterministic parameter.

Suppose γX and γY represent the optimal thresholds for quantizing X and Y respec-

tively, then it is trivial that

J(ψ(X, γX), Y ; θ) > J(ψ(X, γY ), Y ; θ).

On the other hand, Lemma 4.3 gives

J(ψ(X, γY ), Y ; θ) > J(X,ψ(Y, γY ); θ).

The above two inequalities together imply that

J(U(X), Y ; θ) > J(X, V (Y ); θ), ∀ρ,

where U(X) = ψ(X, γX), and V (Y ) = ψ(Y, γY ).

The results on the preferred communication direction is summarized in the next theo-

rem.

Theorem 1. Let θ be a parameter in a tandem network under bivariate additive Gaussian

noises. With single-bit quantizers, the best strategy is to always quantize the observation

with worse noise, and choose the better one as the fusion center.

46

σ2

2 (dB)

-20 -15 -10 -5 0 5 10 15 20

PC

RLB

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7ρ = 0, quantize Xρ = 0, quantize Yρ = -0.5, quantize Xρ = -0.5, quantize Yρ = 0.5, quantize Xρ = 0.5, quantize Y

σ1

2 = 0dB

Fig. 4.4: PCRLB comparison with ρ = −0.5, 0, 0.5

We provide visualization of the PCRLB under different correlation coefficients in Fig. 4.4.

In this figure, σ21 , the noise variance atX , is fixed at 0dB, while σ2

2 varies from−20 to 20dB,

and quantization is imposed on either X or Y . By comparing each pair of curves with the

same ρ, one can clearly see that when σ22 < σ2

1 , quantizing X yields smaller MSE, while

the reverse is true with σ22 > σ2

1 .

Notice that the curve of ρ = 0.5 and quantize Y increases and then starts to decrease at

a particular point σ22 = 6dB. The reason is that in the case σ1 < σ2 and quantize Y, at the

point ρ = σ1σ2

(σ21 = 0dB and σ2

2 = 6dB implies that σ1σ2

= 0.5 = ρ), there forms a Markov

chain θ − X − Y − V (Y ), which implies that the quantized observation V (Y ) is totally

redundant. It can be seen from (4.11) that it reduces to 0 when ρ takes that particular value.

4.3.3 Data Dependency and Redundancy

As illustrated in the centralized estimation example, data dependency could either imply

redundancy or be exploited for better estimation. We now examine the same problem in a

decentralized system.

47

ρ-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

Fis

he

r In

form

atio

n

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

J(U;θ)J(U;θ|Y)

σ1

2 = 4, σ

2

2 = 1

Fig. 4.5: Comparison between J(U(X); θ) and J(U(X); θ|Y )

Similar to the centralized case, to evaluate data redundancy, we need to compare J(U(X); θ|Y )

and J(U(X); θ), where

J(U(X); θ|Y ) =

∫ ∞−∞

∫ ∞−∞

e−θ2

2σ2√2πσ2

· e− (y−θ)2

2σ22√2πσ2

2

·(ρσ1−σ2)2

2π(1−ρ2)σ21σ

22· e−(−θ−σ1σ2

ρ(y−θ)√

1−ρ2σ1

)2

Q

(−θ−σ1

σ2ρ(y−θ)√

1−ρ2σ1

)(1−Q

(−θ−σ1

σ2ρ(y−θ)√

1−ρ2σ1

))dydθ. (4.15)

and J(U ; θ) = J(U ; θ|Y )∣∣ρ=0

. The numerical comparison is given in Fig. 4.5. As with

the centralized case, negative correlation always benefits the estimation performance. For

positive correlated noises, small correlation implies redundancy whereas large correlation

can be exploited to enhance the estimation performance.

We now analytically establish the existence of different parameter regimes in terms of

the correlation coefficient. As the closed-form expression for J(U ; θ|Y ) is intractable, we

resort to a pair of upper and lower bounds.

48

Proposition 1. J(U(X); θ|Y ) is upper bounded by

J(U(X); θ|Y ) <2(σ2 − ρσ1)2

π(1− ρ2)σ21σ

22

, Ju(ρ). (4.16)

Proof. Directly apply Lemma 4.2.

Proposition 2. J(U(X); θ|Y ) is lower bounded by

J(U(X); θ|Y ) >2(σ2 − ρσ1)2σ1

πσ21σ

22

√(1− ρ2)((1 + ρ2)σ2

1 + 2σ2), Jl(ρ), (4.17)

Proof. Use the fact

η(t) =e−t

2

Q(t)(1−Q(t))≥ 4e−t

2

.

It can be verified that both Ju(ρ) and Jl(ρ) decrease monotonically from ρ = −1 to σ2σ1

,

and increase from σ2σ1

to ρ = 1, while achieving identical minimum 0 at σ2σ1

. Also, we notice

that

limρ→±1

Ju(ρ) = limρ→±1

Jl(ρ) =∞, σ1 > σ2.

Hence, J(U ; θ|Y ) is also unbounded at the two extreme points. Since the bounds have fi-

nite values at ρ = 0, J(U ; θ) is bounded, which means that there must be a boundary point

ρ∗ such that for any ρ > ρ∗, J(U ; θ|Y ) > J(U ; θ). Furthermore, it implies that negative

correlation always yields higher FI than its independent counterpart, as does positive cor-

relation beyond the boundary point. Between 0 and that boundary point, data correlation

leads to redundancy, which negatively affects the estimation performance.

It is not surprising that at the two extreme points, the FI is unbounded. However, it does

not necessarily imply perfect estimation at these extreme points. The reason is because the

PCRLB is not tight for the case with quantized observations. To verify this, we compute the

MMSE achieved by the conditional mean E[θ|u(x), y]. In this case, the MMSE estimator

49

ρ-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

PCRLB

Lower bound for PCRLB

Upper bound for PCRLB

MSE

σ2 = 1, σ

1

2 = 4, σ

2

2 = 1

Fig. 4.6: CRLB, its bounds and MSE of the Decentralized Estimation Problem

is computed to be

θ(U = 0, y) =σ2

σ2 + σ22

y − σ2σ2(σ2 − ρσ1)√2π(σ2 + σ2

2)2D

e−(γ−κy)2

2D

1−Q(γ−κy√D

) ,θ(U = 1, y) =

σ2

σ2 + σ22

y +σ2σ2(σ2 − ρσ1)√2π(σ2 + σ2

2)2D

e−(γ−κy)2

2D

Q(γ−κy√D

) , (4.18)

where D =(1−ρ2)σ2

1σ22+σ2σ2

1+σ2σ22−2ρσ1σ2σ2

σ2+σ22

, and κ = σ2+ρσ1σ2σ2+σ2

2. The first term σ2y

σ2+σ22

is the

optimal estimator by using only Y , and the second term can be viewed as a correction term

that takes U(X) into consideration.

In Fig. 4.6, we plot the numerically computed PCRLB, along with its lower and upper

bounds derived in (4.16) and (4.17). Also plotted is the MMSE obtained using the above

estimator. Clearly, the PCRLB is no longer tight in the correlation regimes close to the two

extreme points, i.e., ρ = ±1. This is seen from the divergence between the MMSE and the

numerically computed PCRLB.

Since in the decentralized estimation case, the PCRLB is not tight especially at the

50

γ-5 -4 -3 -2 -1 0 1 2 3 4 5

MM

SE

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

ρ = -0.98ρ = -0.5ρ = 0ρ = 0.5ρ = 0.8ρ = 0.98

σ1

2 = 4, σ

2

2 = 1

Fig. 4.7: MSE vs. γ for different ρ

extreme points, people may question that whether the optimal quantizer threshold and the

preferred communication direction concluded from investigating PCRLB are truly valid.

In Fig. 4.7, numerical results of MSE at the FC with different correlation coefficients are

displayed. One can easily see that the MSE, as same as the PCRLB in Fig. 4.3, is always

minimized at γ = 0. Meanwhile, in Fig. 4.8, the simulations of MSE with different power

ratios are visualized. By comparing it to Fig. 4.4, one can observe the same conclusion that

the preferred communication direction is always from the node with the lower SNR to the

higher one regardless of the correlation coefficient.

Fig. 4.9 illustrates the MMSE under different SNR at the worse node (σ21), with fixed

SNR at the fusion center (σ22). One can see that as σ2

1/σ22 becomes larger, even small

positive correlation can be exploited for better estimation performance. This is because the

particular point ρ = σ2σ1

, at which the MSE reaches its maximum is closer to 0 when σ21

increases.

51

σ2

2 (dB)

-20 -15 -10 -5 0 5 10 15 20

MM

SE

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ρ = 0, quantize Xρ = 0, quantize Yρ = -0.5, quantize Xρ = -0.5, quantize Yρ = 0.5, quantize Xρ = 0.5, quantize Y

σ1

2 = 0dB

Fig. 4.8: Estimation MSE comparison with ρ = −0.5, 0, 0.5

ρ-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

MS

E

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

σ1

2=4

σ1

2=20

σ1

2=50

σ2=1, σ

2

2=1

Fig. 4.9: Estimation MSE with different SNR at the worse node

52

4.4 Summary

In this chapter, we focused on decentralized estimation in a two-node tandem network with

correlated Gaussian noises. The objective is to strive for a better understanding of the effect

of data correlation on the estimation performance. With the Gaussian model, we first es-

tablished the optimality of single threshold quantizer on local observations in maximizing

the FI at the fusion center. This enables us to determine the optimal communication direc-

tion, which is from the node with lower SNR to the other one. Finally, different correlation

regimes are characterized that have different ramifications with regard to their impacts on

the estimation performance compared with that of independent observations.

A natural extension is to study a system involving more than two nodes. For the cen-

tralized case, the observation now becomes

y = θ + w, w ∼ N (0,Σ),

where Σ is the covariance matrix of the additive Gaussian noise. The joint FI to be

J(Y; θ) = −E[∂2

∂θ2log p(y|θ)

]=∑i,j

σij, (4.19)

where σi,j stands for the (i, j)-th entry of Σ−1. Clearly, for a general covariance matrix, the

study of the impact of data correlation on the estimation performance is rather cumbersome.

However, for special covariance matrix structures where correlation can be quantified using

a single or a few parameters, systematic study is indeed possible. One such example is

that all variables have identical pairwise correlation coefficient ρ. For this special case,

similar results can be obtained that parallels the bivariate Gaussian noise case for a two-

node system. The result is provided in Appendix D.

53

CHAPTER 5

ROOM SHAPE RECOVERY VIA A

SINGLE MOBILE ACOUSTIC SENSOR

WITHOUT TRAJECTORY INFORMATION

This chapter focus on the indoor room shape recovery using a single mobile acoustic sensor.

In the problem settings, a polygonal room and a mobile node with co-located loudspeaker

and microphone are employed, and only the first-order echoes are used to reconstruct a 2-D

room geometry.

The room shape recovery problem has been extensively examined in the literature (see

[14–19] and references therein). However, in those works, either a loudspeaker or micro-

phone array is used, or high-order echoes are required; while in this section, the proposed

algorithm uses only first-order echoes collected from a co-located loudspeaker/microphone

sensor for room shape reconstruction. The reason that only first-order echoes are used

is largely driven by practical considerations: higher-order echoes are often significantly

weaker and of poor resolution, leading to unreliable time of arrival estimate. On the other

hand, instead of assuming a single measurement point, we assume a mobile node (e.g., a

mobile phone) that may take measurements at different locations. We show that in the ab-

54

sence of any knowledge of the measurement locations, first-order echoes are sufficient to

recover a wide class of polygonal room shapes. We also investigate its limitation and iden-

tify the class of room shapes that are impossible to recover with the proposed system. This

points to further research direction where additional information/measurements, such as

the trajectory of the moving sensor, are required, which will discussed in the next chapter.

5.1 Problem Formulation

5.1.1 Image Source Model

The basic technique employed in this chapter is the classic model in acoustics/optics,

namely the image source model (ISM) [54–57]. This model converts a source inside a

room into multiple ones outside the boundaries [55], each corresponding to an image of the

original one. These are referred to as the image sources. The model is sketched in Fig.5.1,

in which the polygon is encompassed by four edges W1-W4, and the source is located at

O. Then S1-S4 are called first-order images (e.g. the reflective path O→R3→O), while

S12 and S21 are two examples of second-order images, indicating that they are the images

over two distinct boundaries (the reflective path O→R12→R21→O). The ISM helps us

characterize the room impulse response (RIR), which we discuss below.

5.1.2 Mapping between ISM and RIR

In this section, we introduce the signal model and formulate the problem of room shape

reconstruction in a 2-D space. Suppose the room to be measured is a polygon with N

walls, denoted as Wj, j = 1, · · · , N . Let us consider a pair of co-located loudspeaker and

microphone that moves on a set of points {Oi}Si=1, where each Oi is located at (xi, yi). At

each point, a probing signal s(t) is transmitted from the loudspeaker, resulting in a series

of echoes within the room. Define the ideal room impulse response at a particular point Oi

55

x

y

O

W1

W2

W3

W4

S1

S2

S3

R3

S4 S12

S21

R12

R21

Fig. 5.1: Image Source Model

to be

hi(t) =N∑n=0

ai,n · δ(t− ti,n), (5.1)

where ai,n refers to the attenuation of the direct path (n = 0) or the reflective path between

Oi and Si,n, an image source of Oi, and the number of echoes N is generally greater than

the number of walls N , as there may be second or even higher-order echoes involved. The

parameters of interest in this model are the elapsed times ti,n’s, i.e. the TOAs, and they can

be obtained from the received signal ri(t), which is characterized by ri(t) = s(t) ∗ hi(t).

Notice that for room shape recovery, the TOAs, not the exact form of hi(t) is needed.

As such, broadband acoustic signals such as chirp signals are often candidate waveforms

because of its good time domain resolution.

Assume that the speed of sound is known and invariant, one can map the TOAs with

the room geometry. For the first-order echoes, denote the speed of sound as c and then the

distance from Oi to Wj isc · ti,j

2. (5.2)

In [18], it was shown that the first-order TOAs measured at one single point is not enough to

56

uniquely recover a room, but combined first- and second-order measurements are sufficient.

On the other hand, room shapes can be recovered solely based on the first-order echoes if

multiple distributed microphones are used [15]. In this chapter, we demonstrate that by

using a co-located and mobile loudspeaker and microphone, only first-order echoes are

needed to reconstruct a class of 2-D room shape.

Echo labeling is an important issue that refers to the mapping between the RIR and

room geometry, e.g., the k-th TOA in two different RIRs measured at two separate points

may not correspond to the same wall. To simplify the problem, we assume that the mea-

surement point moves within a substantially small range compared to the room size, such

that the echoes arrive in the same order for all points.

5.2 Room Shape Reconstruction

In this section, we first establish that first-order TOAs are sufficient to retrieve a triangular

room if three distinct sets of RIRs are provided. A simple algorithm to recover triangu-

lar room shapes is provided, which is subsequently extended to cope with more general

polygonal shapes. Limitations of the proposed approach is also identified; we show that

the first-order echo measurements alone has inherent limitation in dealing with parallelo-

grams.

5.2.1 Identifiability by First-Order TOAs

We begin with a lemma that states that triangles can be recovered via three sets of first-order

RIRs measured at three distinct locations without any information about the geometry of

the three measurement points.

Lemma 5.1. For any given triangle T , suppose there are three interior points that are not

on a straight line. Let R be the distance matrix with Rij being the distance between the

i-th point and the j-th edge. Then R uniquely defines the triangle with probability 1.

57

Proof. The detailed proof is given in Appendix E.

5.2.2 Algorithm for Triangle Reconstruction with Only First-order

TOAs

The construction algorithm utilizes the distance matrix. For the sake of simplicity, let W1

be on the x-axis, and one end of it be the origin. We denote the column vectors ci and

rj as the i-th column and the j-th row of the transposed distance matrix RT . Under these

settings, the two other walls are represented as two lines:

W2 : y = a2 · x+ b2

W3 : y = a3 · x.

The intercept of W3 is 0 as it passes through the origin, and for each point, we have yi =

Ri1. Hence, we can map the first-order TOAs to the room geometry through six equations:

|a2xi −Ri1 + b2|√a2

2 + 1= Ri2, i = 1, 2, 3 (5.3)

|a3xi −Ri1|√a2

3 + 1= Ri3, i = 1, 2, 3 (5.4)

The solutions to (5.3) and (5.4) are:

a3 = ±√(

2BC

D2 −B2 − C2

)− 1, (5.5)

a2 =Ba3

B + kC√a2

3 + 1, (5.6)

b2 =kA√a2

3 + 1

B + kC√a2

3 + 1, (5.7)

xi =Ri1 + k

√a2

3 + 1 ·Ri3

a3

(5.8)

where k = sign(a3). Let [u1,u2,u3] = [c2, c3, c1]T−RT , thenA = rT2 (r1×r3),B = rT2 u1,

58

C = rT2 u3 and D = rT3 u1.

From (5.5)-(5.8) we see that there are two sets of possible solutions; choosing the cor-

rect one depends on the angle formed by W1 and W3, i.e., whether it is sharp or obtuse. It

is not hard to discriminate between them, since the incorrect set of recovered points and

edges are not able to reproduce the true distance matrix. For example, assume a3 is actually

negative, then in the set with a3 > 0, the recovered distance between the reconstructed i-th

point and W2 is denoted as

Di2 =(A+BRi3 − CRi1)

√a2

3 + 1√(Ba3)2 +

(B + C

√a2

3 + 1)2. (5.9)

However, the true distance Ri2 satisfies

Ri2 = − (A+BRi3 − CRi1)√a2

3 + 1√(Ba3)2 +

(B − C

√a2

3 + 1)2, (5.10)

which is not equal to Di2. Hence, we have the following algorithm to recover a triangle.

Algorithm 5.1 Triangle Recovery1: set k = 12: calculate a3, a2, b2 and xi using (5.5)-(5.8)3: calculate d2 = [D12, D22, D32]T using (5.9)4: if d2 6= r2 then5: set k = −1 and go to step 26: end if

5.2.3 Polygon Reconstruction by First-Order TOAs

We have shown that first-order RIRs are sufficient to recover triangular room shapes. Un-

fortunately, the result does not hold for arbitrary polygonal shapes. In particular, we demon-

strate below that first-order RIRs alone are not sufficient to recover parallelograms.

59

Lemma 5.2. For any given rectangleR, there is an infinite number of parallelograms hav-

ing the same distance matrix R with R, regardless of the number of measurement points.

Proof. Suppose the edges of a rectangleR are counter-clockwise ordered as y = 0, x = L,

y = H and x = 0. Then the first-order distance vectors must satisfy the following set of

conditions:

r1 + r3 = H1, (5.11)

r2 + r4 = L1, (5.12)

where 1 is an all one column vector. It is easy to find a parallelogram enclosed by four

lines: y= 0, y= ax − sign(a)√

1 + a2L, y=H and y= ax, such that its first-order TOAs

also satisfy conditions (5.11) and (5.12). Since the slope a is an arbitrary real number, there

are an infinite number of parallelograms that can not be distinguished fromR by first-order

TOAs.

If the room shape is truly a rectangle, then Algorithm 1 can be easily adopted to recover

the room shape. Suppose we randomly select three edges as W ′1, W ′

2 and W ′3, then there

must be a pair of parallel edges, therefore two cases can happen: (1) W ′1 is parallel to W ′

2,

and (2) W ′2 is parallel to W ′

3. Note that we do not allow W ′1 parallel to W ′

3 because the

algorithm assumes that W ′1 intersects W ′

3 at the origin (if a3 = 0 is recovered, we can

rearrange the vectors). Both cases are recovered by the algorithm as a shape with one

perpendicular and two parallel edges, and the fourth one will later be recovered to enclose

the shape as a rectangle.

The above reconstruction, however is still subject to the inherent ambiguity between

the rectangle and any of the parallelograms that has the same R matrix. The following

corollary gives a positive result that shows that the first-order RIRs are sufficient to recover

a wide class of 2-D room shapes.

60

Corollary 5.1. For any polygon, if there is a subset of its edges forming a triangle, then it

can be correctly and uniquely recovered solely based on its first-order RIRs.

Proof. The proof is straightforward. Algorithm 5.1 fails only when the selected three edges

can not form a triangle, i.e., two edges are parallel. Conversely, if there is a triangle, then

we can correctly recover it, as well as the remaining edges.

5.2.4 Algorithm of Polygon Reconstruction with Higher-order TOAs

Suppose there are only first-order TOAs included in R, we can reconstruct the polygon

by repeatedly applying Algorithm 5.1 until all rj are examined. Specifically, we apply

Algorithm 5.1 on the starting subset {r1, r2, r3} and keep the recovered a3 and xi’s. Then

we apply Algorithm 5.1 on {r1, rk, r3} for all k = 4, · · · , N , while we use the recorded

a3 and xi’s to compute (ak, bk). Finally, we need to check if the recovered distances dk =

[D1k, D2k, D3k]T match the true distances rk. If not, we need to rearrange the vectors of

R and start over again; this is necessary even for certain shapes that contain at least two

parallel edges such as a regular hexagon. The algorithm will fail only if the room shape is

a parallelogram. However, the R matrix needs to contain only that of first-order echoes,

thus the algorithm needs to be modified if higher-order echoes are included in the distance

matrix.

Instead of R, we use R to denote the distance matrix generated from the received

echoes, which may contain higher-order TOAs, and its dimension is 3 × N . Let us first

consider applying Algorithm 5.1 on {r1, r2, r3}. There are three different cases.

• {r1, r2, r3} does not result in a valid triangle

This happens when some subset of {r1, r2, r3} does not correspond to first-order

echos, then (5.5) may not have a real number solution. This is easy to fix: one can

simply rearrange R until a triangle can be formed.

• {r1, r2, r3} results in parallel edges or a valid but wrong triangle

61

If any two vectors in {r1, r2, r3} correspond to parallel edges, such as in a regular

hexagon, Algorithm 5.1 may mistakenly recover a right angle. Also, if any vector in

the set does not correspond to a first-order echo, the recovered a3 and xi’s are wrong.

In both cases, the subsequently recovered (ak, bk) are also wrong; nevertheless, if an

rk corresponds to an edge parallel to any one in {r1, r2, r3}, the recovered distance

still match the true distances, i.e. dk = rk.

• {r1, r2, r3} results in a correct triangle

If all vectors in {r1, r2, r3} come from first-order echoes without any parallel edges,

then the rk’s correspond to first-order echoes will generate (ak, bk) such that dk = rk,

but higher-order rk’s will not.

The discussion suggests that we need to pre-process the distance matrix R to pair all

parallel edges and separate them into two sets, then recover (ak, bk) as mentioned above,

and finally do a distance mismatch check to eliminate the higher-order TOAs and verify the

reconstruction. Given the outline of algorithm, we can see that the polygons with more than

three non-parallel edges and those with more than two pairs of parallel edges are able to be

uniquely reconstructed. For the first case, if {r1, r2, r3} is correctly selected (all come from

first-order echoes), there is at least one rk for k = 4, · · · , N which can pass the distance

check to validate the starting subset; otherwise, no rk can pass the check. Similarly, in the

second case, more than two pairs of parallel edges guarantees a correct starting subset, and

it follows the same reason as the first case. However, if a polygon has one or two pairs of

parallel edges, it may not be correctly recovered due to inherent limitation of the problem

settings. We have the following lemma to identify the polygons which can be uniquely

recovered.

Lemma 5.3. Suppose a polygon has N edges, including M pairs of parallel ones. The

polygon can be correctly and uniquely recovered either when N −M ≥ 4, or M ≥ 3 if

only first-order information are entirely available.

62

Proof. The first condition generalizes the first case discussed above. Since we only take

one of the two ‘parallel’ sets into account, then recovering the polygon is equivalent to

recovering another one with N − M non-parallel edges. As we have explained that a

polygon with more than three non-parallel edges can be recovered, N − M ≥ 4 is a

sufficient condition for recoverability. The second condition is straightforward by seeing

that at least three pairs of parallel edges guarantees a correct starting set when the algorithm

is applied.

Given Lemma 5.3, the polygon that satisfies neither one of the two conditions has

M = 2, N = 4 or 5, or M = 1, N = 4, in another word, the trapezoid, parallelo-

gram and parallelogram with one corner cut by a fifth edge can not be recovered by only

using the first-order echoes. Consider that non-rectangular parallelogram-shaped room is

very rare in practical, one can directly apply the algorithm to recover a rectangle. How-

ever, additional information, such as the geometry of the measurement points, is required

to recover trapezoids, which will be discussed in the next chapter.

Another special case is thatM = 0, N = 3, i.e., a triangle. We have shown in Sec. 5.2.2

that if only first-order TOAs are included in the distance matrix R, then a triangle can

be correctly recovered with probability 1. However, if high-order information is present,

choosing a starting set {r1, r2, r3} and then verifying other rk by distance match check will

lead the algorithm into an infinite loop. This can be seen from the fact that even the starting

set is correctly chosen, no other rk can pass the following distance match check, and the

algorithm will automatically reject the current combination. The same happens to the case

that the starting set is not correctly chosen.

We have the algorithm given in Algorithm 6.1. Some numerical results are illustrated

in Fig. 5.2. From the figures we can see that the reconstruction is subject to rotation and

reflection ambiguity, and this is predictable as the coordinates of the measurement points

are unknown.

63

Algorithm 5.2 Polygon Recovery1: pair all ‘parallel’ vectors, put them into two separate sets V1 and V2, and the rest in V3

2: rearrange r: put V1 in the front, and V2 in the end3: if M == 0 or M ≥ 3 then4: apply Algorithm 5.1 on {r1, r2, r3}, record a3 and xi5: apply Algorithm 5.1 on {r1, rk, r3}, check if dk = rk6: if dk 6= r for all k then7: rearrange V3 and go to step 48: else9: done.

10: end if11: else12: apply step 4-10 on V1 ∪ V3,13: if step 12 is done then14: recover the edges corresponding to V2, done.15: else(repeats more than a maximum number of times)16: if M == 1 then17: additional information is required.18: else(M == 2)19: recover a rectangle, done.20: end if21: end if22: end if

64

-5 0 5 10-5

0

5

10

-5 0 5 10-5

0

5

10

(a) (b)

-5 0 5 10-5

0

5

10

-5 0 5 10-5

0

5

10

(c) (d)

Fig. 5.2: original (blue) and recovered (dash purple/black) polygons: (a) anirregular quadrilateral, (b) a rotation, (c) a reflection, (d) a hexagon

65

5.3 Summary

While the proposed algorithm suffices to recover a wide class of a polygonal shapes through

only first-order RIRs, additional information or measurement will be needed in order for

the algorithm to work for all possible polygonal shapes. Restricting to only first-order

RIRs, a natural direction is to add additional information with regard to the measurement

points so that, together with the first-order RIRs, unique recovery (subject to reflection and

rotation ambiguity) is guaranteed for all polygonal shapes. Specifically, we will show in

the next chapter that with partial information about the trajectory of the moving sensor, all

convex polygonal shapes can be recovered.

66

CHAPTER 6

ROOM SHAPE RECOVERY VIA A

SINGLE MOBILE ACOUSTIC SENSOR

WITH TRAJECTORY INFORMATION

In the previous chapter, we propose an algorithm to recover 2-D convex polygonal room

shapes using a single mobile sensor with co-located loudspeaker and microphone. In the

proposed algorithm, the mobile acoustic sensor need to move around the room and measure

the TOAs at least at three distinct and non-collinear locations. Only the entire set of the

first-order TOAs are required, while no information about the sensor trajectory, such as

moving path length and the turning angle, is assumed to be known. It has shown in the last

chapter that when the entire first-order and a subset of higher-order TOAs are included in

the distance matrix used to recover the room shape, the proposed algorithm can not cope

with triangle, trapezoid, parallelogram, and parallelogram with one corner cut by a fifth

edge.

In the present chapter, we still consider the problem of reconstructing 2-D room shape

with only the first-order echoes, but with information about the trajectory of the moving

sensor, i.e., the geometry of the measurement points. Given the full geometry of the mea-

67

surement points, i.e., the distance between each pair, it is straightforward to establish that

first order echoes are sufficient to recover any convex polygonal shapes. This however

is a very strong assumption that often requires human intervention, e.g., actually measur-

ing distances between measurement points. However, many mobile devices are capable

of measuring its own path length when moving from one point to another. This chapter

investigates the possibility of reconstructing 2-D room shape with path length of consec-

utive points measured by the mobile device. This weaker assumption, compared with the

knowledge of complete geometry of measurement points, makes it feasible to achieve au-

tonomous room shape recovery.

The present chapter establishes that the above approach is indeed feasible. That is,

given the knowledge of path lengths between consecutive measurement points, one can

recover arbitrary convex room shape by using only first-order echoes collected at three non-

collinear points in the room. Algorithmic procedure that handles measurement noise and

the presence of higher-order echoes is also proposed and experimental results are presented

to validate the approach.

6.1 System Model

The basic model used in the present chapter is the same as the last one, say, the ISM. We

repeat some important notations in the last chapter as follows. We use R to denote the

distance matrix, where Rij represents the distance between the i-th measurement point and

the j-th wall, and ci and rj are the i-th column and the j-th row of RT , respectively. In the

experiment, we denote the number of echoes received at the i-th measurement point as Ni.

6.1.1 Geometry

Consider a convex planar K-polygon. As shown in Fig. 6.1, a mobile device with co-

located microphone and loudspeaker emits pulses and receives echoes at {Oi}3i=1. Without

68

1O

2O

3O

12d

jq1 jR

3 jR 2 jRjW

x23d

p j-

Fig. 6.1: A mobile device is employed to measure the geometry of a room.The mobile device collects echoes at O1, O2 and O3 successively. The dis-tances between the consecutive measurement points are d12 and d23

loss of generality, we assume that O1 is the origin, O2 lies on the x-axis, and O3 lies above

the x-axis. Let ϕ = (π−∠O1O2O3) ∈ (0, π) and the length ofO1O2 andO2O3 be denoted

by d12 and d23, respectively.1 The angle between the x-axis and the normal vector of the

j-th wall is denoted by θj .

From Fig. 6.1, it is straightforward to show that

(R2j −R1j) + d12 cos θj = 0, (6.1)

and

d23 cos(θj − ϕ) + (R3j −R2j) = 0. (6.2)

If however the Rij’s contain higher-order echoes, with probability 1, (6.1) and (6.2) do not

hold simultaneously.

Clearly given d12 and d23 and if ϕ can also be estimated from the measurement data,

then the geometry of the measurement points can be recovered. In this case, first-order

echoes are sufficient to recover room geometry.

1If π ∈ (0, 2π), i.e. we do not have control of where to place O3, then the reconstruction is subject toreflection ambiguity (c.f. Theorem 6.1).

69

6.2 Recovery with Known Distances and Unknown Path

Direction

If for all measurement points, the one-to-one mapping fi : {Rij}Kj=1 7→ ci is known, then

{θj}Kj=1 can be obtained by (6.1) and the room shape is determined. However since echoes

may arrive in different orders at different Oi’s and ci may contain higher-order echoes if

Ni > K, fi is unknown. Then θj’s are also unknown. Therefore we need a way to both

rule out higher-order echoes and find the correct combination of the first-order echoes. We

can then estimate θj’s and the room shape.

Define αjj′ = −R2j−R1j′

d12and βjj′ = −R3j′−R2j

d23. For simplicity we denote αjj and βjj

by αj and βj , respectively. From (6.1) and (6.2), we have

θj = ± arccosαj and θj − ϕ = ± arccos βj, (6.3)

Thus, there are four possible sign combinations for a given j,

θj = arccosαj and θj − ϕ = arccos βj (6.4)

θj = arccosαj and θj − ϕ = − arccos βj (6.5)

θj = − arccosαj and θj − ϕ = arccos βj (6.6)

θj = − arccosαj and θj − ϕ = − arccos βj. (6.7)

We first give the definition of feasibility as follows.

Definition 1. Given a room R and a location O, we say O is feasible if the co-located

device at O can receive all the first-order echoes of a signal emitted at O.

Then we have the following lemmas stating the identifiability of convex polygons using

first-order echoes. We discuss the cases of grouped (echoes for every measurement points

are sorted in a same order with respect to the walls) and ungrouped echoes, separately.

70

Lemma 6.1. Suppose O1, O2 and O3 are feasible and not collinear. Given the correct echo

combination, with probability 1, there exist exactly two sign combinations such that (6.1)

and (6.2) hold simultaneously for all j’s if ϕ and the direction of both−−−→O1O2 and

−−−→O2O3 are

randomly chosen. The two possible sign combinations have opposite signs for ϕ and all

θj’s and correspond to reflection of each other.

Proof. Assume that the ground truth of the polygon is (6.4) for all j ∈ {1, . . . , K}. Note

that (6.4) implies that (6.7) holds for θ′j = −θj and ϕ′ = −ϕ < 0 for all j, which is the

reflection of the room.

Suppose multiple sign combinations hold for a wall. Without loss of generality, let

j = 1. From (6.4) we have

ϕ = arccosα1 − arccos β1. (6.8)

Assume that one of the following equations also holds,

ϕ = − arccosα1 − arccos β1 (6.9)

ϕ = arccosα1 + arccos β1 (6.10)

ϕ = − arccosα1 + arccos β1. (6.11)

Then we have the following three cases:

1. If (6.8) and (6.9) hold, we must have θ1 = 0 which implies thatO1O2 is perpendicular

to the first wall, and ϕ = − arccos β1.

2. If (6.8) and (6.10) hold, we must have arccos β1 = 0, which implies that O2O3 is

perpendicular to the first wall.

3. If (6.8) and (6.11) hold, we must have ϕ = 0, which contradict with the assumption

that O1, O2 and O3 are not collinear.

71

With probability 1, the first two cases do not occur since both ϕ and directions of−−−→O1O2

and−−−→O2O3 are randomly chosen.

If a subset of (6.5)-(6.7) holds for j and j′ simultaneously, then we must have (θj, θj′) ∈

{θj = 0, θj = ϕ, ϕ = 0} × {θj′ = 0, θj′ = ϕ, ϕ = 0}, which again, do not occur due to

randomly chosen measurement points. Similarly, it can be shown that for more than two

walls, (6.4) would imply none of (6.5)-(6.7) holds for all walls.

Lemma 6.2. Given incorrect echo combinations, with probability 1, there exists no solution

to (6.1) and (6.2).

Proof. We illustrate the proof by considering only the case of K = 4. The result can be

easily extended to K = 3 and K > 4.

We assume that the ground truth is (6.4) for all j. First consider parallelograms. The

distances between Oi (i = 1, 2, 3) and the four walls satisfy

R11 +R12 = R21 +R22 = R31 +R32 = a, (6.12)

and

R13 +R14 = R23 +R24 = R33 +R34 = b. (6.13)

We can see that for certain echo combinations, pairs of {αjj′ , βjj′}, j, j′ ∈ {1, 2, 3, 4} are

dependent. Consider for example the echo combination resulting in {α12, α21, α34, α43}

and {β12, β21, β34, β43}. Since α12+α21 = 0, α34+α43 = 0, β12+β21 = 0 and β34+β43 = 0,

we have

arccos(α21) = π ± arccos(α12)

arccos(α43) = π ± arccos(α34)

arccos(β21) = π ± arccos(β12)

arccos(β43) = π ± arccos(β34).

72

Thus (6.3) reduces to two equations

ϕ = ± arccos(α12)± arccos(β12)

ϕ = ± arccos(α34)± arccos(β34).

With probability 1, these two equations do not hold simultaneously as α12, β12 are inde-

pendent of α34, β34 due to randomly chosen measurement points. Other possible cases of

incorrect combination always have at least two equations with independent choice of α and

β. Hence no solution can be found for those instances.

Suppose a combination of echoes is chosen such that we have αjj′ and βjj′′ (i 6= i′, i 6=

i′′). For rooms with no more than one pair of parallel walls, almost surely no echo combi-

nation other than the correct one can make (6.4) holds for all j. This is because for those

rooms, at least one of (6.12) and (6.13) does not hold. Thus some αjj′’s and βjj′′’s are not

dependent since R1j′ , R2j and R3j′′ are randomly chosen from c1, c2 and c3, respectively.

Therefore only the correct combination of first-order echoes can satisfy (6.4) for all

walls.

Note that the distances between the device and both the ceiling and the floor can be

ruled out by Lemma 6.2.

Given Lemma 6.1 and Lemma 6.2, we have the following result on the identifiability of

any convex polygonal room by using only first-order echoes.

Theorem 6.1. One can recover, with probability 1, any convex planar K-polygon subject

to reflection ambiguity, by using the first-order echoes received at three random points in

the feasible region, with known d12 and d23 and unknown ϕ ∈ (0, 2π).

Remark: The room shape is subject to reflection ambiguity for ϕ ∈ (0, 2π). If, however,

we can limit ϕ ∈ (0, π), the recovered shape is unique, i.e. not subject to ambiguity.

In the presence of noise, however, ci is subject to measurement errors. Hence ϕ solved

73

from (6.3) for different j’s are not identical. A straightforward practical algorithm that

handles the measurement errors is given below:

Algorithm 6.1 All Convex Polygon Recovery

1: set N = min {N1, N2, N3}2: for K = 3 : N do3: choose K entries from each of ci, i = 1, 2, 3

4: for k = 1 : . . . ,(NK

)3(K!)2 do

5: for j = 1 : K do6: compute ϕkj = ± arccosαj ± arccos βj7: end for8: end for9: choose and record the echo combination with the smallest variance of ϕkj for the

given K10: end for11: choose the recorded echo combination with the largest K with the variance of ϕkj less

than a threshold12: estimate θj’s using the obtained combination of echoes and reconstruct the polygon

The following lemma states that the knowledge of both d12 and d23 is necessary for

reconstructing any convex polygons by first-order echoes.

Lemma 6.3. If either d12 or d23 is missing, then 1) parallelogram can not be reconstructed.

2) Non-parallelogram can be reconstructed subject to reflection ambiguity.

The proof of the part 1) is to construct a counterexample while 2) can be established in

a manner similar to that of Lemma 6.2.

6.3 Experimental result

In this section, we discuss our experiments conducted in a shoebox classroom.

6.3.1 Experiment Setup

We use a laptop as a microphone and a HTC M8 phone as our loudspeaker. As the loud-

speaker of the cell phone is not omnidirectional and power limited, we place the speaker

74

of the cell phone towards each wall to ensure the corresponding first-order echo is strong

enough. Note that the loudspeaker will record first-order as well as some higher-order ones.

5.025 5.03 5.035 5.04 5.045 5.05 5.055 5.06 5.065

x 105

−1000

−500

0

500

1000

Correlator output at O1 (first direction)

LOS componentwall 2

wall 1

Fig. 6.2: Correlator output at O1 towards the first wall. Peaks with solidellipses correspond to true walls. Peaks with dash ellipses correspond to eithernoise or higher-order echoes

6.3.2 Signal Type

A chirp signal linearly sweeping from 30Hz to 8kHz is emitted by the cell phone. The

sample rate at the receiver is 96kHz. It has been shown in [58, 59] that if the input chirp

signal is correlated with its windowed version, the output may resemble a delta function.

Our simulation and experiment results show that the candidate distances are obtained by

correlating the received signals with its triangularly windowed version outperforms the

correlator output using the original one. Fig. 6.2 is a sample path of the correlator output

collected in the room where this experiment is conducted.

6.3.3 Room Shape Reconstruction Experiment

The proposed approach is verified by experiment in which d12 and d23 are measured by

tape measure. Even if some elements of ci have measurement errors up to 10cm, the room

75

2

O

1

O

3

O

3.66

3.05

1.44 4.60

1.46

3.07

3.65

4.63

1

3

5

º

1

3

6

.

3

º

y

x

1.62

1.97

26.7º

115.2º

-63.5º

-154.6º

Fig. 6.3: Comparison between the ground truth (black underlined) and exper-iment result (red)

can be recovered with negligible error by the proposed algorithm if the subset of ci cor-

responding to first-order echoes are known. In the presence of higher-order echoes, the

proposed algorithm performs poorly when the variance criterion is the only criterion used

to determine the correct combination of echoes. Since most rooms are regular, we add a

heuristic constraint: all the angles of two adjacent walls are between 30◦ and 150◦. An

interesting phenomenon is that sometimes the proposed algorithm is unable to provide the

correct room shape, but ϕ is always estimated close to the true value. Therefore, one can

use the algorithm in Sec. 6.2 to obtain ϕ and then reconstruct the room shape indepen-

dently with full knowledge of the geometry information of the measurement points. The

comparison between the reconstruction result and the ground truth is illustrated in Fig. 6.3.

6.4 Summary

In this chapter, we make progress in room shape reconstruction using only first-order

echoes. Specifically, we established that given partial information about the sensor trajec-

76

tory, i.e., the distances between consecutive measurement points, any 2-D convex polygon

can be reconstructed. In the presence of noise, a simple algorithm is devised that is effective

in recovering the room shape even if the higher-order echoes are present.

However, it can be seen that the time complexity of the proposed algorithm is shown

to be O(∑N

k=3 k(Nk

)3(k!)2

), implying that it is an NP hard problem. Therefore, there are

further research can be done to rule out a large part of the echo combinations to reduce the

complexity of the problem.

77

CHAPTER 7

CONCLUSION AND FUTURE RESEARCH

DIRECTIONS

7.1 Concluding Remarks

In nowadays communication and electronic systems, multi-functional sensors are widely

used. They are able to sense and collect information from the surrounding environment,

perform relatively simple computation locally, and also collaborate with each other utiliz-

ing their communication capability. In this thesis, we addressed problems in two separate

types of applications of such sensors. In the first part, we investigate the decentralized in-

ference with dependent observations in wireless sensor networks. Two different problems

are studied in this part, one is about the cooperative spectrum sensing system using multiple

sensing nodes, and the other one is regarding a decentralized estimation system with corre-

lated Gaussian noises. In the second part, a single mobile sensor bundled with co-located

microphone and loudspeaker is used to recover 2-D convex polygonal room shape based

on the acoustic response of the room.

Chapter 3 is dedicated to study the energy detection in spectrum sensing system. In

this chapter, the optimality (or sub-optimality) of energy detection in two types of spec-

78

trum sensing is examined. For the single node system, the energy detector is proved to

be optimal for most signaling, while for the only case of not being optimal, it is shown to

be substantially close to the theoretically optimal one. For cooperative spectrum sensing

case, a recently proposed framework for distributed inference with dependent observations

(reviewed in Chapter 2) is employed to help establish the optimality of the energy detector

for some signaling, while in other cases, the problem remains difficult to solve.

Chapter 4 is focused on decentralized estimation with correlated Gaussian noises in a

tandem network. The impact of noise correlation on decentralized estimation performance

is investigated. For a tandem network, the optimality of SFQ in the sense of maximizing FI

is established; furthermore, with correlated Gaussian noises, the SFQ reduces to threshold

quantizer on local observations, which is subsequently optimal. Additionally, it is shown

to be better off quantizing the worse node and having the FC at the better node for all

correlation regimes. Finally, different correlation regimes in terms of their impact on the

decentralized estimation performance are identified. The regimes include the well known

cases of negatively and highly positively correlated noise benefiting estimation due to noise

cancellation, and another positive correlation regime which induces redundancy.

Chapter 5 considers a practical problem in room shape recovery using only first-order

echoes. In this problem, a single mobile sensor, equipped with co-located loudspeaker and

microphone, as well as a possible internal motion sensors, is deployed and TOAs are mea-

sured at multiple locations to recover the room shape. In this chapter, no motion sensor is

assumed, thus no trajectory information about the mobile sensor is available. The unique-

ness of the mapping between the first-order TOAs and the room shape is identified when

at least three distinct non-collinear measurement locations are used. It is also shown that

in the absence of higher-order TOAs, all convex polygonal shapes except for parallelogram

can be uniquely and correctly recovered the proposed algorithm, while more information

is required to tackle the parallelogram case.

79

Chapter 6 addresses the same problem with that in Chapter 5, however, partial trajectory

information of the mobile sensor is accessible as the motion sensors are able to measure the

path length between consecutive measurement locations. In this case, the mapping between

the first-order TOAs and any convex polygonal room shape is proved, and an algorithm is

proposed to reconstruct room shape in the presence of higher-order echoes and noise.

7.2 Future Research Directions

We conclude this thesis by listing a few future research directions.

• For energy detection in cooperative spectrum sensing, it is possible that energy de-

tector is optimum for more general cases, e.g., the detection of Gaussian signal in

Gaussian channels with more than one sample. This, however, requires generaliza-

tion of the framework developed in [20] to more general hidden variables. Addi-

tionally, other more realistic channel fading models can be considered, including

non-Rayleigh fading channels, as well as slow fading channels (i.e., correlation of

fading channels in time).

• For the room shape reconstruction with partial trajectory information of the mobile

sensor, the proposed algorithm is able to deal with any convex polygonal shape, with

the expense of time complexity as high as O(∑N

k=3 k(Nk

)3(k!)2

). This is a high

time complexity that sensors, or even personal computers, typically can not process

locally in a tolerable time. Therefore, it is certainly worth improving the algorithm

time complexity by excluding as many as possible the candidate shapes in advance.

80

APPENDIX A

DISTRIBUTION OF PSK AND QAM

SIGNALS OVER RAYLEIGH FADING

CHANNEL

In the Rayleigh fading channel, the complex fading coefficient h can be written in a polar

form as h = |h| · ejθh . It is complex Gaussian distributed with the variance σ2, i.e.

p(h) =1

πσ2e−|h|2

σ2 ,

which means that the real part R{h} , hr and the imaginary part I{h} , hr are both

Gaussian distributed with zero-mean and variance σ2

2, and they are independent.

For a particular PSK symbol sm, the output signal through the Rayleigh fading channel

is h · sm, then it can be further rewritten as |h| · ej(θh+θm), where θm is a deterministic

quantity. Then it is not difficult to verify that the real and imaginary part of h · sm are also

independent and Gaussian distributed. Furthermore, since |h · sm| = |h|, the faded signal

h · sm has exactly the same distribution as h. Therefore, the output signal x(n) = h(n)s(n)

81

has the distribution

p(x) =∑m

πmp(h · sm) =1

πσ2e−|h|2

σ2 ,

which is also exactly identical with that of the fading h.

For a particular QAM symbol, similar to the PSK case, the distribution of h · sm =

|h|rm · ej(θh+θm) is complex Gaussian with zero-mean and scaled variance r2mσ

2. What

makes it different from PSK is that rm are not identical for all constellation points, so the

distribution of the output signal x(n) = h(n) ∗ s(n) is

p(x) =∑m

πmp(h · sm) =∑m

1

πr2mσ

2e− |h|

2

r2mσ2 ,

which is a mixed Gaussian.

82

APPENDIX B

PROOF OF LEMMA 4.3

Our goal is to prove that for any γ ∈ R,

J(ψ(X, γ); θ|Y ) + J(Y ; θ) > J(X; θ) + J(ψ(Y, γ); θ|X). (B.1)

With the Gaussian model for θ and the additive noises, (B.1) can be expanded into (B.2).

1

σ22

+(1− σ1

σ2ρ)2

2π(1− ρ2)σ21

∫ ∞−∞

f(θ)

∫ ∞−∞

e− (y−θ)2

2σ22√2πσ2

2

e−(γ−θ−σ1σ2

ρ(y−θ)√

1−ρ2σ1

)2

Q

(γ−θ−σ1

σ2ρ(y−θ)√

1−ρ2σ1

)(1−Q

(γ−θ−σ1

σ2ρ(y−θ)√

1−ρ2σ1

))dydθ

>1

σ21

+(1− σ2

σ1ρ)2

2π(1− ρ2)σ22

∫ ∞−∞

f(θ)

∫ ∞−∞

e− (x−θ)2

2σ21√2πσ2

1

e−(γ−θ−σ2σ1

ρ(x−θ)√

1−ρ2σ2

)2

Q

(γ−θ−σ2

σ1ρ(x−θ)√

1−ρ2σ2

)(1−Q

(γ−θ−σ2

σ1ρ(x−θ)√

1−ρ2σ2

))dxdθ.

(B.2)

83

Proof. Rewrite the two conditional FIs as:

J1 , J(ψ(X, γ); θ|Y )

t= y−θσ2= C1

∫ ∞−∞

f(θ)

∫ ∞−∞

φ(t) · η(γ − θ − σ1ρt√

1− ρ2σ1

)dtdθ,

J2 , J(ψ(Y, γ); θ|X)

t=x−θσ1= C2

∫ ∞−∞

f(θ)

∫ ∞−∞

φ(t) · η(γ − θ − σ2ρt√

1− ρ2σ2

)dtdθ,

where C1 =(1−σ1

σ2ρ)2

2π(1−ρ2)σ21, C2 =

(1−σ2σ1ρ)2

2π(1−ρ2)σ22, f(θ) is the pdf of the parameter θ, and φ(·) is the

pdf of standard normal distribution. Thus it suffices to prove J2 − J1 <1σ22− 1

σ21.

Using the fact that max η(t) = 4, one can further bound the difference of the two

conditional FIs as in (B.3).

J2 − J1 = (C2 − C1)

∫ ∞−∞

f(θ)

∫ ∞−∞

φ(t) · η(γ − θ − σ2ρt√

1− ρ2σ2

)dtdθ

+ C1

∫ ∞−∞

f(θ)

∫ ∞−∞

φ(t) ·(η

(γ − θ − σ2ρt√

1− ρ2σ2

)− η

(γ − θ − σ1ρt√

1− ρ2σ1

))dtdθ

=1

2π

(1

σ22

− 1

σ21

)∫ ∞−∞

f(θ)

∫ ∞−∞

φ(t) · η(γ − θ − σ2ρt√

1− ρ2σ2

)dtdθ

+ C1

∫ ∞−∞

f(θ)

∫ ∞−∞

φ(t) ·(η

(γ − θ − σ2ρt√

1− ρ2σ2

)− η

(γ − θ − σ1ρt√

1− ρ2σ1

))dtdθ

<2

π

(1

σ22

− 1

σ21

)+ C1

∫ ∞−∞

f(θ)

∫ ∞−∞

φ(t)

(η

(θ − γ + σ2ρt√

1− ρ2σ2

)− η

(θ − γ + σ1ρt√

1− ρ2σ1

))dtdθ.

(B.3)

The task is now equivalent to showing that the second term in (B.3) is non-positive. Let

a1 ,θ−γ√1−ρ2σ1

, a2 ,θ−γ√1−ρ2σ2

, s , ρt√1−ρ2

. The inner integral in (B.3) is

√1− ρ2

|ρ| ·∫ ∞−∞

e− (1−ρ2)s2

2ρ2

√2π

· (η(a2 + s)− η(a1 + s)) ds (B.4)

84

Define

q(s) = η(a2 + s)− η(a1 + s).

We can show that q(s) is an odd function at s = −a1+a22

, i.e. q(s − a1+a22

) = −q(−s −a1+a2

2). Without loss of generality, consider θ > γ, which implies a2 > a1 > 0. Consider

the following different regions of s in terms of the effect on the sign of q(s).

• s < −a2, then a1 +s < a2 +s < 0. In this region, η(t) is a monotonically increasing

function, thus

q(s) = η(a2 + s)− η(a1 + s) > 0.

• −a2 < s < −a1+a22

. This condition implies that a1 + s < a1 − a1+a22

< 0 and

0 < a2 + s < a2 − a1+a22

, thus

η(a2 + s)(1)> η(a2 −

a1 + a2

2)

= η(a1 −a1 + a2

2)

(2)> η(a1 + s).

The inequality (1) holds as η(t) decreases monotonically when t > 0 and similarly

for (2). The equality follows as η(t) is symmetric at t = 0. Therefore, in this region,

q(s) = η(a2 + s)− η(a1 + s) > 0.

Thus q(s) > 0 when s < −a1+a22

. Due to the odd symmetry of q(s) about −a1+a22

, we can

easily obtain that q(s) < 0 when s > −a1+a22

.

If θ < γ, then a2 < a1 < 0, the discussions are similar and the reverse inequality holds.

The results can be summarized as follows

q(s) ·(s+

a1 + a2

2

)< 0, for θ > γ,

q(s) ·(s+

a1 + a2

2

)> 0, for θ < γ.

85

√1− ρ2

|ρ|

∫ −a1+a22

−∞

e− (1−ρ2)s2

2ρ2

√2π

q(s) +

∫ +∞

−a1+a22

e− (1−ρ2)s2

2ρ2

√2π

q(s)

ds

(1)=

√1− ρ2

√2π|ρ|

(∫ −a1+a22

−∞e− (1−ρ2)s2

2ρ2 q(s)ds+

∫ −a1+a22

−∞e− (1−ρ2)(−(a1+a2)−r)

2

2ρ2 q(−(a1 + a2)− r)dr)

s=r=

√1− ρ2

√2π|ρ|

(∫ −a1+a22

−∞

(e− (1−ρ2)s2

2ρ2 − e−(1−ρ2)(−(a1+a2)−s)

2

2ρ2

)q(s)ds

). (B.5)

Therefore, we can split the integral in (B.4) into two parts and rearrange the terms in (B.5).

In (B.5), the first equation follows by changing the variable r = −(a1 + a2) − s. We now

analyze the integral in (B.5) for two cases:

• θ − γ > 0.

As s < −a1+a22

in the integral in (B.5), q(s) is always positive as discussed before.

Meanwhile, it is not difficult to see that s2 > (−(a1 + a2) − s)2, thus e−(1−ρ2)s2

2ρ2 −

e− (1−ρ2)(−(a1+a2)−s)

2

2ρ2 < 0. Hence, the integral is negative.

• θ − γ < 0.

Under this condition, q(s) is always negative as discussed. Moreover, as a2 <

a1 < 0 when θ − γ < 0, we have s2 < (−(a1 + a2) − s)2, and thus e−(1−ρ2)s2

2ρ2 −

e− (1−ρ2)(−(a1+a2)−s)

2

2ρ2 > 0. Hence the integral is also negative.

Therefore, the integral is always less than 0. Plug this result back into (B.3), the difference

of the two conditional FI’s is bounded by:

J2 − J1 <2

π

(1

σ22

− 1

σ21

)<

1

σ22

− 1

σ21

.

This proves (B.1) hence

J(ψ(X, γ), Y ; θ) > J(X,ψ(Y, γ); θ).

86

Although we assume that the parameter θ is Gaussian distributed, the entire proof is in-

dependent of the distribution of θ. Hence, for arbitrary θ, within the framework of single

threshold quantizer, it is always better to impose it at the sensor with the worse noise,

though the single threshold quantizer is not always optimal for arbitrary parameter.

87

APPENDIX C

PROOF OF LEMMA 4.2

To prove lemma 4.2, we only need to show that

1

p(t)= et

2

Q(t)(1−Q(t))

is a monotonic increasing function over (0,∞), i.e.,

d

dtet

2

Q(t)(1−Q(t))

=et2

(2tQ(t)(1−Q(t))− 1√

2π(1− 2Q(t))e−

t2

2

)> 0. (C.1)

Since it is continuous at t = 1, we can prove it separately on two sets, (1,∞) and [0, 1].

• Monotonicity of p(t) over (1,∞)

We only need to show that h(t) = et2Q(t) is monotonically increasing function on

(1,∞), i.e., h′(t) > 0, as (1−Q(t)) increases monotonically in this region. To prove

it, we need to use an inequality

Q(t) >t√

2π(1 + t2)· e− t

2

2 . (C.2)

88

Taking derivative of h(t), we have

h′(t) = 2tet2

Q(t)− et2 1√2πe−

t2

2

(1)>

et2

2√2π

(2t2

1 + t2− 1

)> 0, ∀t > 1.

The inequality (1) follows Eq. (C.2). Hence, p(t) decreases over (1,∞).

• Monotonicity of p(t) over [0, 1]

Observe Eq. (C.1), it is a quadratic form of Q(t), though the coefficients are depen-

dent on t. We can construct a function

f(t) = −2t · x2 +

(2t+

2√2πe−

t2

2

)· x− 1√

2πe−

t2

2 .

Then the roots of f(t) = 0 are

x1(t) =1

2−

√2πte

t2

2

2(√

2πt2et2 + 1 + 1),

x2(t) =1

2+

√2πte

t2

2

2(√

2πt2et2 + 1− 1).

If we can prove that x1(t) ≤ Q(t) ≤ x2(t) over (0, 1), then the inequality in Eq. (C.1)

is satisfied. First, since 0 ≤ Q(t) ≤ 12

over (0, 1), then Q(t) ≤ x2(t) always holds.

To show x1(t) ≤ Q(t), we first check two boundary points: x1(0) = 0.5 = Q(0),

and x1(1) = 0.1066 < 0.1587 = Q(1). The derivative of r(t) , Q(t) − x1(t) is

given as

r′(t) =− 1√2πe−

t2

2 +

√2π

2

(1 + t2)et2

2√2πt2et2 + 1(

√2πt2et2 + 1 + 1)

,

89

so we can check the derivative on the two boundary points:

r′(t)|t=0 = − 1√2π

+

√2π

2> 0,

r′(t)|t=1 = − 1√2πe(√

2πe+ 1)< 0,

then there must be at least one t0 such that r′(t0) = 0. Furthermore, if there is only

one such t0, combined with r(0) = 0, r(1) > 0, it is easy to show that r(t) > 0 over

(0, 1). Identifying t0 is equivalent to look for the root of the following equation:

πet2 · (t2 − 1)2 = 2. (C.3)

Let x , t2, and define g(x) = πex · (x− 1)2. Then we have

g(0) = π > 2,

g(1) = 0 < 2,

g′(x) = πe2 · (x2 − 1) < 0, x ∈ (0, 1),

which implies that there exists one and only one x0 ∈ (0, 1) such that g(x0) = 2,

and consequently one and only one t0 ∈ (0, 1) such that r′(t0) = 0, which means

r(t) = Q(t)− x1(t) > 0, t ∈ [0, 1].

Overall, the funtion p(t) is monotonically decreasing over (0,∞), and increasing over

(−∞, 0), where t∗ = 0 gives its maximum p∗ = 4. Also, limt→±∞ p(t) = 0, so-called the

bell-shaped function.

90

APPENDIX D

CENTRALIZED ESTIMATION FOR A

MULTI-NODE SYSTEM IN GAUSSIAN

NOISES

Extension to the multi-sensor scenario is feasible under certain symmetric assumption of

the correlation matrix which is included below. While this does not cover all possible

covariance matrices of observations, the intention is to provide insight on how various

correlation regimes might impact estimation performance.

Consider the case of a multi-node system with multivariate additive Gaussian noises

that have identical pairwise correlation coefficient. The sensor observations can thus be

written as

y = θ + w, w ∼ N (0,Σ),

where Σ is the covariance matrix taking on the following special structure

Σ =

σ2

1 · · · ρσ1σn... . . . ...

ρσnσ1 · · · σ2n

.

91

It is a straightforward exercise to show that the joint Fisher information is

J(Y; θ) = −E[∂2

∂θ2log p(y|θ)

]=∑i,j

σij,

where σi,j stands for the (i, j)-th entry of Σ−1, the inverse of the covariance matrix Σ:

Σ−1=

1+(n−2)ρ

(1−ρ)(1+(n−1)ρ)σ21· · · −ρ

(1−ρ)(1+(n−1)ρ)σ1σn

... . . . ...

−ρ(1−ρ)(1+(n−1)ρ)σnσ1

· · · 1+(n−2)ρ(1−ρ)(1+(n−1)ρ)σ2

n

.

Therefore the joint FI can be expressed as

J(Y; θ) =1 + (n− 2)ρ

(1− ρ)(1 + (n− 1)ρ)·

n∑i=1

1

σ2i

− ρ

(1− ρ)(1 + (n− 1)ρ)·

n∑i=1

∑j 6=i

1

σiσj.

The contrasting case is when the noises are mutually independent, and the corresponding

Fisher information is

J(U ; θ|Y )∣∣∣ρ=0

=n∑i=1

1

σ2i

.

This serves as the baseline to study the impact of noise correlation on the estimation per-

formance. Thus, the problem becomes comparing the following quantity to 0

d , J(Y; θ)− J(Y; θ)∣∣∣ρ=0

=(n− 1)ρ2

∑ni=1

1σ2i− ρ∑n

i=1

∑j 6=i

1σiσj

(1− ρ)(1 + (n− 1)ρ),

and then one can find the different regimes of ρ that has distinct impact on the estimation

performance.

• ρ = − 1n−1

. This value leads to a rank deficient covariance matrix, which means

that in this case, the noises are linearly dependent. Therefore, perfect estimation can

be achieved by complete noise cancellation, i.e., by a linear combiner with properly

chosen coefficients.

92

• − 1n−1

< ρ < 0. This is the regime of ρ such that data correlation may benefit

estimation performance through partial noise cancellation. It is comparable to the

negative correlation regime in bivariate Gaussian scenario.

• 0 < ρ <

∑ni=1

∑j 6=i

1σiσj

(n−1)∑ni=1

1

σ2i

. In this regime d is negative, which means noise correlation

leads to redundancy. Additionally,∑ni=1

∑j 6=i

1σiσj

(n−1)∑ni=1

1

σ2i

is the boundary point defined in a

similar manner to that of the bivariate Gaussian case.

•∑ni=1

∑j 6=i

1σiσj

(n−1)∑ni=1

1

σ2i

≤ ρ < 1. Positive high correlation is beneficial to estimation as it does

for the bivariate case. In this case, noise cancellation more than compensating for

signal power reduction.

• ρ = 1. Similar to the discussions in bivariate Gaussian noise scenario, perfect esti-

mation can be achieved at this value if noise variance are not identical to each other.

Thus the study of the bivariate case (i.e., a two-node system) generalizes to the multi-variate

case. Similar observations can be made for the decentralized system yet the analysis is

rather cumbersome and one has to largely resort to numerical evaluation.

93

APPENDIX E

PROOF OF LEMMA 5.1

Proof. As dipicted in Fig. E.1(a), the sides of T are counter-clockwisely labeled. Without

any loss of generality, we fix O1 to be the origin, and the first side of the triangle to be

y = −r11. To check if there is any triangle other than T such that there exist three points

O′i satisfying r′i = ri, we need to consider two cases: one side is tilted and two sides are

tilted. We will consider them seperately.

1. One side is tilted

Suppose W3 stays intact and W2 is being tilted, then O2 has to keep at its original

position, i.e. O′2 = O2. As shown in Fig. E.1(a), observe that W2 is an external

common tangent of the two circles centered at O1 and O2 with radii r12 and r22,

it is easy to show that the other external common tangent W ′2 is the only candidate

satisfying r′i = ri, i = 1, 2. Therefore, two measure points are not enough to uniquely

define a triangle. However, to keep r′3 = r3, O′3 has to be on the line O1O2. Hence,

for three noncollinear points inside T , there is no triangle remaining ri while having

only one side different from T .

2. Two sides are tilted

In this case, both W2 and W3 are being tilted. Now randomly fix W ′3 to be any

94

x

y

O1r11

r12

r13

O2

O3

W1

W2W3

W ′2

x

y

O1

O′2

O′3

W1

W ′3

W ′2W ′′

2

r13

r22

r32

(a) (b)

Fig. E.1: (a) one side is tilted, (b) two sides are tilted

line that is r13 apart from O1, and without any loss of generality, we choose it to be

positive sloped and above the origin. Then W ′3 is represented as:

W ′3 : y = a3 · x+ r13 ·

√a2

3 + 1.

Therefore, the other two points O′2 and O′3 can be easily determined as:

O′2 =

((r21 − r11) + (r23 − r13) ·

√a2

3 + 1

a3

, r21 − r11

),

O′3 =

((r31 − r11) + (r33 − r13) ·

√a2

3 + 1

a3

, r31 − r11

).

Now consider the second side W ′2, which can be generally expressed by

W ′2 : a2 · x+ b2 · y + c2 = 0,

95

and the coefficients must satisfy the folllowing equations to keep r2 unchanged:

|c2| = r12 (E.1)∣∣∣∣a2

a3

((r21 − r11) + (r23 − r13)

√a2

3 + 1

)+ b2(r21 − r11) + c2

∣∣∣∣ = r22 (E.2)∣∣∣∣a2

a3

((r31 − r11) + (r33 − r13)

√a2

3 + 1

)+ b2(r31 − r11) + c2

∣∣∣∣ = r32 (E.3)√a2

2 + b22 = 1 (E.4)

Eq. (E.1) indicates that c2 is a constant independent of a2 and b2. As equivalent

to linear equations of (a2, b2), Eq. (E.2) - (E.3) have finite numbers of solutions,

referring to the common tangents of the two circles centered at O′2 and O′3 with radii

r22 and r32. Meanwhile, Eq. (E.4) implies that (a2, b2) must be a point on the unit

circle on the 2-dimensional plane of (a2, b2). Since a3 is randomly selected, then the

intersection of two random lines falls on the unit circle has probability 0, because

the boundary is of measure 0 in a 2-dimensional coordinate system. On the other

hand, if a3 is correctly selected, then there must be a single solution to Eq. (E.1) -

(E.4), which is T . This case is illustrated in Fig. E.1(b), where W ′2 and W ′′

2 are two

common tangents.

In conclusion, any triangle can be exactly recovered by measuring at three noncollinear

points inside it, when first-order information is provided.

96

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VITA

NAME OF AUTHOR: Fangrong Peng

PLACE OF BIRTH: Wuhan, China

DATE OF BIRTH: Nov. 17, 1986

GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:

Syracuse University, Syracuse, NY, United States

Chalmers University of Technology, Gothenburg, Sweden

Huazhong University of Science and Technology, Wuhan, China

DEGREES AWARDED:

M. S., 2014, Syracuse University, Syracuse, NY, United States

M. S., 2010, Chalmers University of Technology, Gothenburg, Sweden

B. S., 2008, Huazhong University of Science and Technology, Wuhan, China